Indeed, but the GPS fix you're getting is 4-dimensional: if you have position, you also have precise time. Therefore - listening for the GPS time is not necessary all the time, just update when a precise-enough fix is achieved.
It's worth noting that to get time with .1s accuracy you don't actually need a GPS fix, just to pick up the 50-bit-per-second navigation signal for a couple seconds, from one satellite.
Based off of some quick and dirty calculations for a GPS satellite at approximately 20,000km above sea level, the difference in distance between a satellite directly overhead and one straight out over the horizon is around 5500km.
That's actually only about 18ms difference in delay. Unless there's something I'm failing to account for here, you should be able to get more than 0.1s precision based off of a single satellite's signal.
Subtract a flat ~75ms to account for the minimum delay (~66ms @ 20,000km) and half of the variable delay, and you shouldn't really be more than about 9ms out.
The distance to an overhead GPS satellite is somewhere in the range of .07-.09 light-seconds. The planet's width is basically irrelevant when your goal is .1 second accuracy.
