Going the "government sponsored program that conducted experiments on inmates" route is an absolute non-starter.
They will probably have to construct a back story wherein Cage is falsely imprisoned in a prison run by a private organization, which of course, is secretly in AIM's pocket.
AIM, of course, is gleefully running experiments (variations of the super-soldier serum) on the inmates. A number of inmates are killed, and these are chalked up to prison violence, riots and the like. Cage is their only success.
So, AIM plans to 'study' him further to try to understand why only his enhancements 'took'. Cage, having other plans, escapes.
AIM's now hunting him down, but he can't go to the authorities since (a) he doesn't have any evidence (other than the fact that bullets don't seem to be able to take him down
) and (b) well, he IS a fugitive.
All this sets up a third act where he takes this particular wing of AIM down and clears his name, possibly with the help of a lawfirm suspiciously named Walters, Murdoch, Weying and Hawk.
I get that AIM are not among Cage's traditional foes, but I think the setup fits within the context of the MCU. What are the Extremis soldiers if not tech-based super soldiers?