If I flash my unit back to base from Abandoned Hall, can I use the Abandoned Hall trigger to buff the unit I flashed back?

No, you cannot. The unit left Abandoned Hall during Flash's resolution, so it is no longer "here" when the Abandoned Hall trigger resolves.

Here's the sequence:

1. Play Flash targeting your unit at Abandoned Hall.
2. Flash resolves — the unit moves from Abandoned Hall to base.
3. Abandoned Hall trigger condition met — because Flash resolved, the "when you play a spell" condition is fulfilled.
4. Trigger placed on the chain — "give a unit you control here +1 [S]."
5. Trigger resolves — you need to choose a unit at Abandoned Hall. Your unit is at base now.

Same principle as the Challenge + Abandoned Hall FAQ (#10497): if the unit leaves Abandoned Hall during the spell's resolution, it's not there to receive the buff when the trigger resolves. Contrast this with Ride the Wind, where the unit arrives at Abandoned Hall during the spell's resolution and is there for the trigger — direction matters here.