If I move off a battlefield after holding, then move another unit into that battlefield later that turn, will it start another showdown?

Yes, moving another unit into that battlefield will start another showdown, even if you have already held and scored at that location earlier in the turn.

According to RiftJudge FAQ #10950, FAQ #10027, and Rule 344, here is why this happens:

1. Losing Control: When you move your last unit out of the battlefield to your base, the battlefield becomes unoccupied and uncontrolled (Rule 323.6).
2. Contesting: When you move a unit back into that battlefield, you are moving into a location that you do not currently control. This makes the battlefield Contested (Rule 445).
3. Showdown Trigger: Because the battlefield is contested and the turn is in a neutral open state, a showdown begins (Rule 344).

Important Nuances:

• Non-Combat: Because the battlefield was unoccupied when the showdown began (assuming your opponent did not also move a unit there), this will be a non-combat showdown. Per Rule 348.2.a, since you are the only player with units present, you will regain control of the battlefield once the showdown closes.
• No Scoring: You will not score another holding point or trigger conquering effects (such as from specific cards) at that battlefield for the rest of the turn, because you have already scored it earlier (Rule 465, FAQ #7447).