Does Improved Logistics allow you to re-use your military power to settle two new worlds in the same turn?
You military in race for the galaxy is never spent. Not even in the two moments between playing two worlds with Improved Logistics.
Even when using Space Mercenaries and paying cards to increase your military, the new military score will apply to both worlds played, and does not need to be divided
The only exception to this is the Imperium Supply Convoy card in the Alien Artifacts expansion, which explicitly calls out that the second world it can occur is with excess military only.
Also, note that you can't use powers from the first world you just settled or conquered to conquer the next one.