There are also quite a few wells, particularly in west Texas, long thought to be played out, that are producing again.....
What you may be thinking of is what is called a stripper well. Usually a well that produces 10 barrels or less per day that is uneconomical to keep online at $10 or $20/barrel but becomes very economical at 80 bucks per. Maintenance on an existing well can be killer when it is required so these marginal producers sometimes get iffy.
Another thing that can be done is horizontal drilling which can also produce and area for a longer period of time but gets very, very pricey to the tune of 30 grand or more per day of drilling costs. Big, big gamble but can extend the years of profitable production for a field.