Local cafes are everywhere.
I've been to some really good ones, but I normally stick to Starbucks and can't honestly recall specific cafes by names.
However, neighborhoods like Hiroo, Sangenjaya, Omotesando and Kichijoji all have local cafes. I think those are neighborhoods that come to mind as far as places with cafes.

As for chain places, Nana's Greentea has been mentioned but I have to say it's decent but not great (maybe a bit better than KyoHayashiya) although I've only had tea drinks there. For tea latte or any tea drinks, it's hard to beat Gion Tsujiri tea at Sky Tree. Otherwise I actually think I'm best off going to Starbucks for maccha latte over Nana's or KyoHayashiya or most certainly Doutor's. But at Starbucks, they can sometimes butcher it even in Jpn and I always ask them to make it foamy a la cappuchino. Japanese ppl buy lattes at konbinis which I think are garbage. On the other hand, drip coffees and iced coffees I find to be pretty high-quality most everywhere including konbinis and Doutor's.