Is it ok to get a bunch of RG-58 sized connectors so that thinner types of coax will fit as well?
Not if you want to do it right. Proper termination of coax requires a smooth transition from the coax shield to the connector. Mechanical compatibility is an issue as well if you don't want your cables constantly breaking.
I've never seen a solder-on SMA connector except for semi-rigid coax. Usually they come in two kinds: clamp and crimp.
An example of the clamp type is Amphenol 901-128-11. This has a clamp nut assembly which the cable passes through, then the shield is folded back upon it. The plug then threads onto the clamp nut, with the folded back shield clamped between the two pieces.
If the coax is too large, it will not pass through the clamp nut. If the coax is too small, it might work well enough. The coax will flop around, which will eventually fatigue the metal in the shield. And the geometry of the connector will be out of specification, compromising high frequency performance. Below 1 GHz you may not notice.
Clamp connectors don't require any special tool to install. That's about the only advantage they have. In an application where the cables are frequently handled, expect them to fail either due to working themselves loose, or from the shield breaking where all the strain is focused at the clamp point.
An example of a crimp connector is Amphenol 901-9511. These connector have an inner ferrule which slips between the connector shield and dielectric. An outer ferrule is then crimped over it, making both a solid mechanical and electrical connection.
If the coax is the wrong size in either direction then the ferrule won't fit at all.
Crimp connectors are the way to go. You'll have to buy a crimp tool, but they're less clunky, and the crimp provides a more solid mechanical connection which takes the strain off the cable braid. They will never come loose. They never fail.