Needing 10cm and having $10 basically makes it a dead end.

With more money, it could have been possible to triangulate the location using piezo transmitters, or cheap 433MHz radios or maybe even the chatter from BlueTooth transmissions.

But there is one more GPS-based option available to you, if you can afford a data communication link too.

Place a fixed GPS receiver + data server somewhere within a couple of km from your measurement location, and then use a mobile phone with GPS + internet access. Let the phone retrieve correction information from the fixed receiver.

If you design the hardware yourself, you could use WLAN or longer-range BlueTooth communication as a back-end channel for the correctional information. Maybe the professional edition of ZigBee.