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For smaller things, a coping saw or fretsaw would definitely be the tool of choice.  That said, when you cannot use them (or a bow saw) due to lack of throat depth or any other reason, a keyhole saw may be the answer.  You would start by drilling a starter hole and then use your key hole saw to saw the hole.  The only limitation on using a key hole saw is the radius of the hole you are trying to cut must be large enough to allow the saw’s use.  Said another way, a keyhole saw, unless quite small, will not work on small radius holes.  You should be able to find a keyhole saw at Home Depot or Lowes for a few bucks – or if you’re adventurous, you can make one.