Yes you can hike out from Bridge Camp. There is a gate at 410 and if the clearcutting there hasn't grown up too much since the last time I was there you can spot where the bridge is by looking down the hill from 410. If I recall it's probably a half mile or less walk (uphill on gravel road).
Oh, and Hancock TImber owns the gate and requires a permit to access their land. You can get a walkin day pass for $8. You may be able to get a drive in permit too but they tend to be spendy.
https://www.hancockrecreationnw.com/white-river/non-motorized-recreation-access/non-motorized-recreation-access-permit-individuals/non-m
I know that Jay Cohen rafted the White below Bridge camp right up to where the low pool mark is behind Mud Mt. back in the early 90s with special permission from the Army Corps and according to him it was the best section of the river by far with several nice rapids. There was hope for a while that this section would become available to the public but that never came to light unfortunately.
Good luck with your adventure-- and yes you should have a very open mind about wood and portaging if you take this on.
Edited by JoesKayak - 07 Aug 2018 at 12:55pm