Yesterday afternoon I (male, late 30) went to the supermarket in my neighborhood to buy something. Since the sky was menacing rain, I took along an umbrella. Lucky me, when I got out from the supermarket it was raining, so I opened the umbrella and headed home.
Few meters ahead I saw a woman (looking about my same age) walking in my same direction without any umbrella. As I got close to her, I covered her with my umbrella and offered her to walk together. She thanked me and stayed under the umbrella, while making some small chat (how long are you being living here, the road works, the rain, etc.).
During the entire walk I tried to keep some distance from her (at least one palm between our elbows), despite the umbrella not being that large, just to avoid any creepy factor.
When we reached the second block (I live on the third) she told me she had to cross the road, because she lived on the west side on the road, while we were walking on the east side, and didn't want to bother me with crossing twice. So she thanked me again and had a short run under the rain to reach the opposite side walk.
I took it as a polite way of hinting me of not following her until her door, so I didn't insist on offering her some more shelter, and kept walking on my side of the road. However, later I reconsidered it as just being polite from her and not wanting to bother me and that I might have come out as rude for just walking away.
How could I have offered her if I could further shelter her from the rain without sounding creepy or pushy?
Generally you shouldn't offer further help. As soon as she wants to go her way you let her do so. You are a random stranger to her, there is every chance she doesn't want to lead you to her home because you might be just a nice guy, but you might be psycho stalker. Any insistence to walk her home might be seen as creepy.
Saying all that the only possible answer to your question in my opinion is - give her the umbrella and go your own way. Might go with a card so she can return it someday. This is the only way you are not going to make her question your intentions. You will end up soaked but with the warm and fuzzy feeling that you did something nice for someone else. Might even get a date afterwards if she decides to call and return it.
I don't think you could have.
Different people have different sensibility to what makes them uncomfortable. I know that, for me (a woman), the mere fact that you propose your umbrella would have made me uncomfortable.
The fact that she accepted your proposition doesn't necessarily mean she is comfortable. I know that I would be capable of accepting this proposition out of fear, even if I don't want to accept.
The fact that she did some small talk tends to indicate that she was comfortable with this situation. But it is not a proof and you can't be certain of that. Even if she laughs, you can't be certain that she is comfortable. I tend to laugh a lot when I am uncomfortable.
All this to say that you couldn't offer further shelter and be certain that she doesn't think you are creepy and pushy.
However, if you still want to ask, you can minimize the risk of sounding creepy/pushy.
First, check if you feel that she is comfortable. Don't ask her but see if she is doing her share in the conversation (and not just responding to your question then maybe asking "and you?").
Then, when she told you she has to cross the road. You can maybe say something like that "I wouldn't have minded but, anyway, have a nice day". With this, you show her that crossing wasn't a problem for you but that you respect her decision (and that you don't expect her to change her mind). If she really wants to use the umbrella and was just being polite, she will ask you to accompany her further.
Finally, do not insist! People who insist are creepy. You don't want to be creepy.
It's quite a good thing you used a "NL" tag, because I live in France and that would definitely sounds like you were hitting on her.
That being said, when I offer my help to someone and they say something like :
It's alright, you can leave my package here! I live just to the opposite side of the road!
I would just ask :
Are you sure? That doesn't bother me at all to help you a little longer!
If they are indeed sure, I will just just let it go, but it's important to let people know they are not asking for too much, and that I truly want to help. They might just say it's alright because they feel like they are bothering me, so they are just trying to be polite. I just want to let them know I did notice they were trying to be polite, but my offer still holds. After that, I consider my job done, if I may say.
Also, context is really important. If it's a girl my age, I will never ever insist, but if we are talking about an old lady, I definitely would (reasonably, of course).
Congrats to your mom though, who raised you so well that you are well-mannered to offer your umbrella, while feeling (slightly?) awkward about it.
I would suggest that changing how you offer the umbrella in the first place would help set people at ease about it.
While not getting wet is appreciated, having someone simply walk over and hold an umbrella over one's head—and necessarily encroaching on personal space—could be a very awkward thing. It puts the person in the position of having to refuse it if it is unwanted, and the people who are most likely to want to refuse are also the most likely to have difficulty doing so.
So consider in the future, walking up to be near the person where you can be seen, but not too near, and simply offering your services in a non-threatening way. How do you do that? Start with "it seems like you're getting pretty wet, there" or perhaps "enjoying getting wet?" (asking a question that the person will say "no" to helps engender a sense of power that will be favorable to you). Keep unexcited, low pitched voice, falling at the end or barely rising if it is a question. Whatever the response, you can get from it an idea on how to proceed. If you're met with a positive response, simply extend your umbrella to your side (not toward the person) as if to make space, and gesture with your hand that the person is welcome to step into that space, perhaps with "I'm headed that way, but I don't mind a detour if you don't object?" The nonverbal part is important, though, and can come first. Nonverbal signals are much easier to get past the ego.
By the person stepping under the umbrella instead of you placing it overhead, he or she will have a deeper sense of control over the situation. Combined with asking questions where "no" is the response that indicates cooperation with you, this is much more likely to make the person feel comfortable accepting further assistance. You didn't force an unwanted situation on someone. Instead, you offered freely-chosen escape from the rain, which if taken, was bodily expressed as wanted (such having a significant psychological effect on the person, as self-chosen physical actions have great power to change cognitive frames).
Add some humor. If the person has smiled or laughed, even "you look like a drowned rat." Act like you don't mind being slightly disagreeable—this can counteract any "creep" vibe from coming across as too desperately agreeable. Act like you don't care one way or the other if the offer is accepted, but also suggest you don't mind going a bit out of your way. Make this offer to help keep the person dry before the person has to indicate that your continued presence is unwanted. Make clear your own path so all the person has to do is nothing to separate. Make each step something the person can willfully choose without having to say something awkward or forceful.
I usually tell the exact location I'm going to, right at the moment I offer to share my umbrella with a woman.
This way, she gets to decide how long we share the umbrella. If she wants to, she can pretend she has to go somewhere else, and she doesn't need to be afraid that I'd follow her right to her doorstep.
I think being physically attractive goes a long way in avoiding being seen as "creepy" in such a situation. While this isn't a variable one can necessarily control, one can attempt to improve physical attractiveness by losing weight, building muscle, wearing nice clothes and being well groomed.