I found the cause of this behavior in the `keymap' section of
(info "(Elisp)Special Properties"):
     The property's value for the character before point applies if it
     is non-`nil' and rear-sticky, and the property's value for the
     character after point applies if it is non-`nil' and front-sticky.

So, I've made changes so as to add the (rear-nonsticky t) property
to MIME buttons.