I find that the 2nd steep can often be shorter, because the tea seems to have fully hydrated at that point, and bitterness can be more concentrated if I let that steep be as long as the first one was. This seems to be most important with senchas and many other green teas, but I have had a couple of white teas that could get bitter, and did the same thing wth them (shorter 2nd brewing). After the 2nd infusion, I again go longer, regardless of the time between infusions.
So for a green tea, I might do 30", 10", 30", 45", 1 min, 90", 2 min.
For a white tea, depending on how I pack the gaiwan or pot, it might be the same, or longer, 60", 20", 2 min, 3 min, etc, but still I shorten that 2nd infusion, same as green tea.
When in doubt, pour out a sip of the 2nd infusion, taste quickly, and decide whether or not to make the infusion longer. That's hard to do with a 10 second infusion window, but easier if your infusion times are 20 seconds or more.