Since it is a reward redemption, no, UA F is your best bet unless you can score NH F.
SQ C is great, but it is impossible to book using miles.
OZ C can be great too, but you'd have to connect at ICN (SFO/LAX-ICN with new C seats)
NH C seat isn't great (still angled lie-flats unless you fly JFK-NRT IIRC)
TG C seat isn't great either (angled lie-flat) and service is inconsistent, not to mention the need to connect at BKK.

Your options would be
1) Try to get NH F
2) Try to get OZ F (via ICN)
3) Try to get NH C (via JFK)
4) Stick with what you have (UA F)
5) Try to get OZ C (new seat only)