bad alternator?

[Stocks10]

Hey guys i've searched and searched and found nothing, hopefully y'all can help me out a tad. I've got a 2004 s10 crew cab LS, i'm trying to figure out why i went out to it one night and when i started it my battery light was on and the voltage gauge was reading 12-13 volts(battery power). pulled the alternator (figured that was what was wrong) took it to oreillys and had it tested before i shelled out the cash for a new on, test came back as a good alternator. pulled the battery had that tested too, also good(after charging). put it all back together and drive it to work once everything is good get in to go home start it and low and behold battery light is on again and voltage gauge is reading battery voltage again. I am pretty much at my wits end trying to figure this out. some added info it started on a day when temps dropped to mid 40s or so and they have just been lower since the problem began. Thanks for any help you guys can give me.

[LoneWolf04]

Wouldn't trust the bench test on the alternator. It does sound like the alternator is going though. Would double check ground wire to the passenger side frame as well as to the engine block. Clean them up if needed. Frame likes to rust and the block likes to get gummed up from oil leaks.