You say "time" but you appear to show voltage. If voltage
ramp time is long then so will be startup time, because you
have a minimum headroom requirement.
This is where a power-on-reset circuit comes in, which
just keeps everything else quiet until reference is up
(or close enough). It can probably be taken off the startup
circuit with a little extra gain / offset / buffering.
If you have that, and you need some "stuff" to receive
-a- (not to say, a -good-) reference, then maybe
analog-mux between the real reference and a dumb
resistor-FET-FET (-FET-FET) stack based on the POR
state. But really, 0.9V of headroom for a 4-FET stack
is not much. To do better incrementally, use wider
devices (lower current density). Or go to a simpler
(likely less accurate / consistent) topology.