Did you ever work in retail? The “star” sale items — usually two or three things — are loss-leaders. They are designed to get you in the store and the ads usually are quite clear that “quantities [are] limited.”
That said, the best tactic is to know-and-go on the day the deal starts. For groceries, it’s usually on a Wednesday. I realize that’s not always possible but if you can make it a short in-and-out trip just to get the specials it might help. It’s up to you to do the ROI for your time to “time it” right.
The other tip is to ask floor/restock staff if or when an additional shipment may come in. I’ve not had much luck asking the “fulfillment” folks; they just pull from their list which already eliminates or substitutes out-of-stock items.
Good luck and it’s nice to see someone watching their wallet.