July 2019
At the moment, whether or not you can pop a syringe after being revived inside the 2nd ring depends upon a combination of the game tick in which you're revived, and your latency to the server.
This is infuriatingly random, frustrating, and really, really stupid.
2nd ring damage should be calibrated such that you either CAN, or CANNOT, pop a syringe in time; not some BS arbitrary limitation based upon uncontrollable factors.
Personally I think 2nd ring damage should not be so high as to preclude reviving inside the ring.
Obviously legendary backpack / lifeline passive / lifeline drone not withstanding.