2000 Tacoma battery light

2000 Tacoma battery light
Recently my battery light came on and then went off over a period of 2 wks. As time went on it came on more frequently and would stay on longer.

Saturday the truck wouldn't start and we replaced the battery which was 5-6yrs old. After replacing the battery, the battery light remained on and did not turn off. It ran well until the next day. I drove it about 40-50 miles during that time and then it wouldn't start. I jump started it and went to drive home. I ended up having to turn my lights off in order to make the 3/4 mile trip home. It died in my drive way.

Monday, I took the alternator out and had it rebuilt for $45. That evening I took the truck to Auto Zone to make sure the battery was fully charged. It was. They tested the alternator and it only showed 24 amps when it should be 80 amps.

Thursday I decided to try the radio out, the battery light came back on. I noticed that when I accelerated the light would turn off.

Now it's Friday and again it won't run unless I get a jump start and then only for a few miles.

I'm thinking the problem is still the alternator. I was told that if I took the alternator out and brought it to Auto Zone, they could get a more accurate result from the test.

If the alternator's good, what other things should I check for? The alternator belt and fuse? The battery cables? What first?