Pu-erh is particularly good for this. A good quality pu-erh can be steeped 10 times or more. So long as you are taking the initial cups early in the day you can keep drinking it way past your 3pm deadline.
Worth investing in, seems expensive but the re-steeping qualities make it very economical. Also, unlike with other high grade loose teas - which take concentration and experience to get right consistently - pu-erh is relatively idiot proof, over steeping and using the wrong temperature rarely makes the brew (and subsequent re-brews) unpalatable.

Also, you get what you pay for, a low quality pu-erh can be rank, a good one... Hard to describe, like the scent of a old leather chair whilst having a fine cigar (and I am not a smoker).

Otherwise, plenty of low grade Japanese teas that are low in caffeine - levels of caffeine in bancha can be very low. I don't go for artificially de-caffeinated options but from the range of natural caffeine free beverages/tisanes I would recommend Barley tea (called mugi cha in Japan) or an infusion from dried mulberry leaves which tastes very similar to some green teas. There's also buckwheat tea and roasted black soy bean tea, both of which can be excellent.