They do, but usually they start filming fall shows in July and they go on for some months (usually they finish before February or March). I think also mid season shows start filming around that month if they have a lot of stuff to do (Smash for example was always a mid season show for both season and they always started filming around late July. They had more stuff than a traditional show, anyway, like recording the songs, learn choreographies, etc), but it can happen that they start in late August.
I learnt all this stuff thanks to GG
New shows only film the pilot before the pick ups, if they're lucky they start filming the other episodes.