The problem with the 58-63 American is the width of the engine bay. A modern four cylinder doesn't even fit good because the intake manifold sticks so far out to the side. I helped put a Pinto 2.3L (or was it 2.0L?) in a 54 Nash Rambler (50-55 Nash Rambler is virtually the same car as the 58-63 Rambler American), but it was an early model Pinto engine. The carb and intake just did clear the left side after a slight bit of "massaging" to the inner fender panel.
The one more or less modern engine that will easily fit is the Ford I-6 with the intake made onto the head. A late 80s 250 would be great! All you'd need to do is fab engine mounts and have a driveshaft made. I have used a hot rod type tubular crossmember to install an engine in an early American -- the rails are 27" apart on the inside (the engine compartment is only 20-1/2" wide at the narrowest spot, right where the suspension mounts). You will have to make a trans crossmember. Those cars have a continuous rail from bumper to bumper, almost like a frame with a floor pan welded to it. I just made a simple angle iron crossmember for the trans and used carriage bolts going through the floor and rails, the rounded heads weren't felt under the carpet. The rails are 18 gauge steel, so not heavy enough to screw into, and the inner rockers are made in such a way that you can't run a bolt sideways through the rail without cutting a big hole the rocker cover. You could probably drill new holes in a wrecking yard rear trans crossmember and mount it the same way. You'll need to swap a trans that will fit the engine you use, of course. The early (60-65?) Falcon and Mustang 200 sixes used the same manual three speed trans as the Rambler (BW T-96), but the 250 uses a SBF bolt pattern and never used that old tranny.
A V-8 or V-6 will fit, but that narrow section makes it tight! The engine compartment does get wider above the upper a-arm mounts though. The main problem with a small V-6 is the accessory mounts, but you could use early hot-rod type mounts to get the accessories up high instead of off to the side. Forget a later model AMC six, they are too long! The compartment is rather short too, all but the small Ford six (144/170/200/250 -- the 240/300 is way too long!), even the Nissan I-6, is 3-4" too long.