I'd double check the chain position by physically looking at it. It may be easy to not see the smallest rear sprocket, and mistake the second sprocket for the first.

There are adjustments that make shifting more precise.

Two little screws on the rear derailleur are high and low limit screws. They are used to keep the derailleur travel from going too far and the chain falling off either end of the cassette (or freewheel). One for high limit, and one for low limit.

There should also be inline cable adjustment somewhere, either at the shifters or at the derailleur or both. You can use those adjusters to effectively make the cable housing longer or shorter, and thus adjust where the derailleur stops with each shift.

Something like a broken spring somewhere could also affect shifting.

Also note, there are a few parts that can slip, usually with a lurching feel, or perhaps spinning the cranks without going anywhere. That would be a different problem and needs to be stated.