I disconnect the ignition before I start my engine after an oil change and I run it on the starter until I see oil pressure.
You can speed up that process by priming the filter before install if it's of the regular canister type.
With the can upright and using a round flat ended drift or simular, gently press back the valve disc a bit (they don't move far) in it through one of the small holes around the big center hole and slowly pour oil into the small hole(s) that have opened (NOT the big center hole, all that'll do is dump out when you go to install the filter) You gotta go slow because the filter element will restrict flow to some degree.
After about 1/3~1/2 quart you can stop and put it on (don't wait too long or oil will seep through the filter element and get into the middle where it'll just dump out as soon as you tip the filter) and the valve disc will prevent the oil from coming back out, just like it does on the engine once it's been run once.
Oil pressure will come up
very quickly compared to just winding on an empty can.
I imagine the same could be done with the older filter version too, but it'd prolly get messy in the process.