This doesn't exactly answer the question, but is related.

During tuning I hit the side edges of the bridge with little fillips to stabilize things.  My idea is that that will cause the bridge to reach a stable position on the belly and cause the strings to be stable in their notches.  Also sometimes I press down equally on the lengths and afterlengths of each string.

Does it do any good?  How would I know?

When I pick up the fiddle, I try to tell---or should I say guess---by ear alone whether any are out of tune.  Then I use the electronic tuner and play all four strings in a row, noting which ones are out of tune and approximately how much.  If any are out of tune, I turn all the fine tuners of the out-of-tune strings the estimated amounts needed---without playing any strings---and give the bridge a little whack from each side with my middle fingernail as mentioned in my post above.  Then I start again---playing all four again to know whether they are all in tune yet. 

So in a sense I tune them all at the same time, not in a sequence.