Well, your insurance policy works by seeding humans across great gulfs of time and distance, insulating them from single-event catastrophes. It seems the parameter to optimize is thus the product or sum of space and time. By that measure, getting to Alpha Centauri faster might not be an actual improvement.
But if we switch to looking at two states in time, humanity at t0 and humanity at t(year), we'd want to maximize the distance between clusters of humans and the total number of seperate clusters. In that case, if we had an improvement in travel velocity, we wouldn't use it to reduce the time, we'd use it to increase the distance.
We'd also not send multiple ships to the same destination, which wouldn't add to the number of seperate clusters, we'd scatter them out to cover as many destination stars as possible.
Since the greatest threat to their long-term survival comes from humanity itself (or its remnants or replacments), it also indicates that many of these ships should go long, go deep, change course multiple times, and run silent, just like they were hiding from an alien species bent on human extermination.
Sensible precaution, perhaps the best way to do this is have the ship loiter at the edge of the solar system, and then suddenly decide to undertake the journey at some random unannounced point in time, and vary the acceleration rate and cut off the acceleration within some interval so the precise velocity position is unknown by outsiders. Probably the vast bulk of humanity would not care after the ship disappears beyond the edge of the solar system. As for potential stars to travel to there is:
Alpha Centauri A G2 V 4.4 light years
Procyon A F5 V 11.4 light years
Tau Ceti G8 V 11.9 light years
Delta Pavonis G6 V 18.6 light years
Eta Cassiopeia A G0 V 19.2 light years
82 Eridani G5 V 20.3 light years
Beta Hydri G1 V 20.5 light years
all approximately yellowish sunlike stars.