Let the auctioneer run the auction. The Official Rules don't specify an auctioneer, but the least conflict of interest, on average, probably arises from having the player who landed on the property, and declined to buy, be the auctioneer. Then run the auction as a proper New York auction - the type everyone is familiar with from movies and TV.
It is the auctioneer's responsibility to set a minimum price by asking if anyone will bid $X (any amount made up by the auctioneer). If there are no takers at that price the minimum bid request is lowered until an opening bid is successfully solicited (essentially a semi-Dutch Auction). This is commonly done by calling out "Do I hear $X" three times before adjusting the solicitation. Then the bidding continues until no further bids can be solicited.
After each bid is recognized by the auctioneer, the auctioneer is responsible for soliciting higher bids that are any desired amount above the currently recognized bid. Bids between the currently recognized amount and the solicited bid amount can only be made once the auctioneer calls out "Going once at $Y. ... Going twice at $Y. ... Going thrice at $Y and SOLD for $Y." A valid intervening bid amount is only valid between utterance of the first "Going" and the "SOLD".
Running the auction as a proper New York Auction allows the auctioneer discretion to run the auction efficiently, so the game is not unduly slowed, without introducing any artificial constraints on the valid bid amounts.