> 9600 baud 8N1.  Looking through the datasheet, I see the TRMT
> bit is set once a transmission is complete, is it safe to assume I
> can begin a new transmission once that bit is set?

Yes. As soon as TRMT is set then Elvis has left the building and
a new value can be loaded

         bank0
         movlw   byte        ;load Transmit reg with data to send
         movwf   txreg

         bank1
         btfss   txsta,trmt   ;byte transmission complete
         goto    $-1
         bank0

As 9600 is relatively slow, the TRMT test loop is quite long, and
you could use that time to do something else (maybe prepare
data for transmission ?), periodically checking for TRMT=1

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