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Did you install new transmission gears or re-use ones already in the case? Had you driven the car before its restoration? Was there 2nd gear noise then? If you re-used the 2nd gear that came with the transmission, chances are it's worn and the teeth edges are rounded or pointed. That will contribute to slipping out of gear, besides worn detents in the shift tower.
Because the Model A's transmission gears are straight-cut, there will be a little "singing" in them no matter what, especially in 1st and 2nd gears. Back in the day when roads were poor and driving speeds were slower than we typically drive our Model A's today, people stayed in second gear for a long time because they often couldn't go fast enough on those bad roads to get into 3rd gear. So, 2nd gear was used for long time stretches at a time. That accelerated wear on that gear over the years. The rounded teeth resulted from harsh shifting, up and down. I would bet that if the gears are original, you'll find the 2nd gear teeth very pointed and probably rust pits on the teeth faces if moisture has taken its toll on the metal in the last 90+ years.
I agree with nkaminar - a transmission rebuild is in your future. If you replace the 2nd gear with a new one, you should spend the extra couple hundred dollars on a new cluster gear. After all, the 2nd gear teeth and the cluster gear teeth have been in contact with each other all these years. If the 2nd gear is so badly worn that it makes especially loud whining noise and jumps out of gear, it's a sure bet the cluster gear is also worn. It'd be a shame to go to all the trouble of replacing only the 2nd gear and then discover after test driving the car that the transmission still makes noise. That would be the cluster gear telling you it should have been replaced, too.
As long as you're sleuthing the problem, follow the guys' advice regarding the shift tower inspection and repair. Now would be the perfect time to eliminate one possible contributing factor in the tranny jumping out of gear. Rebuilding the shift tower won't eliminate the whining 2nd gear nuisance, but at least you'll have peace of mind that the shift tower functions properly and is not the cause of jumping out of gear.
Marshall