Many years ago we had a rat problem, caused when a neighbour had been storing grain in a loose heap in an open-fronted shed. The rats multiplied really quickly, but then once the grain was used, they spread out to all the surrounding farms, so all got his problem.
We used bait boxes carefully positioned, airgun, and the terriers, plus of course the measures you already use to give them nothing to eat. I think of those things, the terriers were the biggest deterrent - even when they didn't kill them, the dogs made life very unpleasant for rats. Now, we occasionally find the dogs have killed a young rat, so presumably one scouting for fresh territory.
Other neighbours used Rentokil for advice and to place bait boxes.