Someone care to explain how precision even could be increased? It doesn't seem possible from what i know of the Bitcoin network.
Everybody upgrades their client to use the new precision.
Since this would be a relatively benign upgrade in the protocol that most people would accept, all it would really take is updating the version number in the protocol.... sort of like the IPv4 vs. IPv6 issue. That would require an update of the clients, but it wouldn't be too bad.
The main issue is that at least for awhile the network would be forked like what happened with the 0.3.10 update, where users running the older software would not be able to recognize the newer blocks, but the older blocks (and transactions) would be recognized for awhile. Doing a shift of that nature would end up having the majority of the network "voting" with their CPU processors, where the largest number of hashes would decide if the change is accepted or not. It doesn't have to be "everybody", but those left behind on the smaller network would be quickly "forced" to make the switch.
If we do make such a switch, I hope it is done thoughtfully and other "bugs" in the protocol are strongly looked at as well.
Considering the substantial overhead in the current protocol, the few extra bytes for additional precision in the transactions is irrelevant and shouldn't even be an issue here, particularly if some of those overhead issues were cleaned up with a tighter protocol.
I don't see a graceful hand-off though, which would be the issue. The main issue is that the "new" transactions that would have the extended precision data would be rejected by older clients on the first block that has one of these "new" precision transactions incorporated into that block.