Nate Nagel wrote:

I'm having a hard time visualizing, probably because I've been up late
every night this week either playing with power tools or huffing flux
fumes. On a typical older stop and waste valve, if I shut the valve
off, can I remove the packing nut and shove some more packing in there
without having the water shut off *prior* to that valve?

I think the answer is "yes", you can shut off the valve and loosen and
remove the packing nut to add valve stem packing. I just had to do that a
few weeks ago with an old toilet supply valve. I just shut it off without
turning off the water supply anywhere else, then loosened the valve stem
nut. In my case, even repacking the valve stem didn't work because it was
an old and corroded valve with a bent valve stem. So I had to replace the
valve after trying the repacking trick first.

The easy way for you to be sure is to turn off the valve and just loosen the
valve stem nut a little and see what happens.