Transmission on Scion slow to shift out of second gear, but only once

I have looked through the questions recommended by SO and none of them seem to address this specific issue – and it’s the only one the car has, otherwise it’s in immaculate condition.

It is a 2007 Scion with automatic transmission. In the morning when I drive it, I’ve just gotten used to the fact that it will only go to second gear for about the first 60 seconds of driving. No problem, I live in a suburban area, just go slowly around 30MPH and eventually it will shift into higher gear. Possibly, maybe, a bit longer on a cold day but not much.

My suspicion is that there is a gummed-up something that releases as soon as the transmission is warm.

My question(s) are, what is the likely problem, is it likely to get worse, and is there any non-invasive approach (like additives) that would be a good first try in order to resolve it? Thanks!