You can get crystal oscillators that have a frequency control signal and are more stable than the run of the mill oscillators. Changing the GPS oscillator would require modifying a very tightly populated circuit board. Perhaps not possible.

What about some of the SDR (software defined radio) projects that aim to
implement GPS functionality?   If you used the GPS chipping rate (1.023 MHz)
to dicipline the 10 MHz oscillator then you are less sensitive to crystal instabilities. You are updating the crystal one million times a second rather than once per second. This is assuming that the chipping rate of the transmitter is just as good as the
1 PPS signal.   This info from here;
https://www.e-education.psu.edu/geog862/node/1753
and here;
https://en.wikipedia.org/wiki/GPS_signals

Even using the 50 bits/sec data rate of the GPS signal would allow updating the
GPSDO faster than the 1 PPS signal.

Pete.
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