I don't think there's any way for female headers that work with the standard square male headers on your Nano to be much more low profile because you're limited by the length of the male pins. You can get Nano clones without the headers soldered. The Pro Mini also comes without headers soldered and that board could be a good choice to use on a finished project where you aren't frequently uploading sketches.
What do you consider expensive / cheap?Wouldn't the male header still be to long, even if you find small and inexpensive female header?maybe just buy longer female headers to gain some extra space to your shield?
I think the best course of action is what Pert suggested - nano or pro mini clone without pre-soldered connectors, and solder on round pin header. That gives you a gap of ~0.2" between the boards. You can also solder a pro mini (but not a nano, since there's stuff on the underside) directly onto your circuit board - put holes matching the pro mini footprint on your PCB, and solder it down. Making the connections can be tricky, but it's not that bad. I position the board, put no-clean gel flux on the holes, and then solder them - when the solder finally wets the metal around the hole in the PCB you're soldering it down to, you'll see the solder suddenly appear to get sucked in - that's what you're going for. Also inspect the board from the bottom to make sure you see solder on the underside of all holes. I've used this method in a number of designs, and it's now my favorite way to mount pro mini's on a larger PCB - though I would hesitate to use it where the board will be exposed to significant vibration or flexing.
Are you thinking about something like this...
You can get the round pin headers on eBay. They're a little more expensive than the square pin ones but still not too bad
Yes, but without the staggered leads. More like this...https://www.mouser.com/ProductDetail/3M-Electronic-Solutions-Division/929870-01-16-RA?qs=%2fha2pyFadujRJI8DsGJUZWlaYrgHz%2fuPzohApSC1ewKo2kmrfa%2fcIQ%3d%3d
If you look closely at the connectors I posted you will see the pin holes go all the way through the connector so as long as you have corresponding holes in the PCB you can plug from either side or allow the long male pins to sit through the PCB.