How do logbook endorsements work?
Well you’ll have to comply with the terms of §61.73 in order to convert your military pilot ratings over to civilian pilot certificates and ratings. You would also need to obtain a logbook endorsement to operate a complex and high-performance airplane legally under the FAA’s rules.  This really should not be challenging for you.  If you really want to rent a twin i.e. a Cessna 310, a Diamond DA 42, etc.,  I would approach your FBO and ask to do a check out in the aircraft that you wish to rent.  The FBO instructor can include both the complex, and if necessary, a high-performance logbook endorsement along with the training.  I don’t know how many hours total that you have - I assume it’s somewhere in the order of 200 or so hours, which s typical of UPT grads - but FBOs may have additional minimum hour requirements to rent said aircraft.  They may also have to consult with their insurance policies to see if such a thing as feasible given how little time you have on light twin airplanes.

Keep in mind here that while you have time in real 'high performance' airplanes, GA aircraft are going to handle differently than what you’d have expected. This is particularly true of light twins which are going to be a lot more treacherous in the event of an engine failure than a centerline thrust jet like a T-38 would be.  I’ve worked as a CFI in a couple of FBOs now and have had the opportunity to check out a number of military personnel on light GA aircraft.  Out of all the people that come through those places, the highest number of accidents on smaller GA aircraft for rent were caused by airline pilots and military pilots simply because they are not used to the feel and control characteristics of smaller, lighter aircraft.  A guy can have 1000 hours in F-22s and still struggle to land a Cessna 172 because the handling characteristics are so different.   Said check out may take several hours both to get familiarized with the handling characteristics of the airplane as well as become familiar with the avionics and operation within the NAS as a civilian pilot.