When the price change is announced, anyone from the original pool has [30] days to match. First to match immediately wins the work from the existing contractor. Loser pays to move their equipment out; winner should have included that cost when they agreed to match.
This system works for two reasons. First, the first bidder will be cautious about requesting increases if it means risking losing the work. Second, the change-over cost, borne by both the outgoing loser and incoming winner, imposes just enough friction to grant the first winner an incumbency bias. This bias helps keep bidding in the original auction competitive.