You do not indicate what type, wattage lamp you are using. You should be able to pull down the trim and perhaps see the brand of the fixture. This might lead you on to get a more modern trim.

Also, most cans have an adjustment for the debth of the socket so you might be able to bring the socket down.

It is unusual to have only one recessed fixture in a room. Could this perhaps be an eye-ball trim?

I'm using a 75 watt incandescent bulb. I pulled out the housing. It says it is for a max of 75 watts, and there is a sticker saying that it's "union made." I couldn't find any maker information -- no serial # or brand name or anything.

It's actually one of several dated-looking recessed lights in the room. The others are 7" squares with covers that hang down diagonally, and I'd like to replace them too.

It's not an eyeball trim, although that would make sense (it's beside the fireplace). It's a very old-fashioned looking flat round trim.

I would love to bring the socket down but I examined it and can't figure out how I could do that. The wiring box is integrated into the top of the canister. The socket appears to be built into the top. Any ideas?

Here is my two cents. To get more light in the room make sure that you are installing the maximum wattage lamp for that recess can which is 75W. I would recommend using a par-30 or par-38 75w lamp make sure it is a flood light. (Not spot) To replace the trim I would remove the existing and go to Home Depot or Lowes and look for a new trim in the style you desire. When you find a trim you like compare it with the existing trim and see if they are the same size and if they connect to the can housing in the same fashion. If they look very similar odds are that the new trim will fit into the existing recess can. If you need to extend the lamp down in the recess light housing there should be a wing nut on that can be loosened for adjustments. If you loosen this wing nut the plate that the light socket sits on should be able to be adjusted up and down in the housing. If the socket is not adjustable I would buy a socked extender also at Home Depot and Lowes and use it. Just make sure that if you save any recipes so if the trim or socked extender did not fit it can be returned.
PS - If the 75w bulb is not bright enough you can install a 100W equivalent CFL light bulb. I know that the 100W is over the max wattage but in actuality the 100W equivalents only use around 23W and operate at a much lower temperature than standard light bulb. The only drawback to CFL bulbs is that the majority of them will not work on a dimmer switch.

Here is a link that might be useful: Recessed Can Light Guide