Different alt texts for the same featured image on different pages

I have a problem with featured images: I use the same featured image on multiple pages but in different contexts. So I want to use different alt text for the same featured image on different pages. It doesn't seem possible to set custom alt text for a featured image. Changing the alt text of a featured image on a page will change the alt text on all pages that use that image as an featured image. Do you know of a workaround to give a featured image an individual alt text that is independent of the alt text in the library?