Can you re-use millitary power to conquer two worlds on the same turn (with Improved Logistics)
Yes
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.