I have tried hosting an image online and then creating a signature with this image pointing at the URL. It displays the first image fine but the second image I upload to overwrite the first, doesn't appear as the first one is being cached. The idea would be to update the image and keep the URL to make it easier for users to not need to keep changing their signature.
Looking for the URL displays the second image fine, it just won't refresh on Outlook.
Most email providers cache images these days, so that's not going to work.
Gmail cache images of the same name since 2013 (https://www.litmus.com/blog/gmail-adds-image-caching-what-you-need-to-know/) and though there might be ways of cache-busting it through adding a query string to the end of the picture: e.g. http://example.com/images/same.png?12345 (HT: https://redant.com.au/how-we-do/cache-busting-gmail-new-image-caching/), that still involves changing the code, so isn't an option for you.
Yahoo has been doing a similar thing from 2018 (https://www.litmus.com/blog/yahoo-mail-introduces-image-caching-what-marketers-must-know/).
Apple Mail recently added Mail Privacy Protection which pre-fetches emails and caches them. Fresh Inbox did some testing and found "If you have the same image across multiple emails (and emails across accounts on the same email client), the image is only fetched once." (https://freshinbox.com/blog/a-technical-take-on-ios15-mail-privacy-protection/)