OK I got the new controller, and I can access players 3 and 4 now, though I did have to change some things in the stock MAME32 UI settings. Initially, when I plugged in the new controller, MAME still wouldn't recognize players 3 and 4. However, I was playing around in the MAME32 UI, and found that in Options->Default Game Properties, on the controllers tab, it showed 4 dropdown boxes for the different game controllers, but only the first two had active selections. I enabled controllers 3 and 4, and now the MAME tab settings would recognize the players 3 and 4 controls. Thanks thomas_surles for the suggestion of joy2key -- I wound up using that to map my player 1 coin and start buttons to "O" and "K" so that I could run the games that say "press OK to continue", without hooking up an actual keyboard.