Can you still use improvised orders after the commander with the upgrade has already activated that turn? The local concurrence is no, but I’m not sure if that’s right.

Improvised Orders can be used at any time.

Given that the card makes no reference to anything like that, the next time this happens you should ask them for a page number from the Rules Reference Guide supporting that assertion.

The use of the card is not an action or a free action. The timing for the use of the card is pretty clear. As arnoldrew said, next time people make a claim like that, ask for where in rules they are basing their claim. Most rules issues like this seem to be a case of over complicating the issue unnecessarily.

Fun fact, if you use it before the unit that has it activates, when that unit activates, you may recover and be able to use improvised orders a second time in the same turn