The play itself does not change, or care if the tire is on or off. But the tire on does give you more leverage as you're testing, so I think that's a good thing.
You can still check with the tire off of course, but it takes more physical effort.
A reading of .001" is tight, but it's within the spec (from .001 to .010 inch) so you are theoretically good. If you want to compare, you could re-do (or just check the other side?) the nuts with the book procedure to see if you get a change.
Only reason I'm saying that is that perhaps with only .001" there is some of that coming from physical flex of things including the mounting of the gauge.
I don't know this, but I do think it's possible.
Whatever is easier though. Either measure the other side to compare, or since you already have this side opened up, take the few minutes to do the adjustment again.
In a pinch, if you think you're measurement is a true indicator of play in the bearing itself, then you could roll with that because it's not bound up. Personally though, I think it's worth a couple of tries this first time with the gauge to see what you can get out of it.
Since mine always fell into the .005-.006 inch category, I'm just naturally leery of such a tight tolerance as one thousandth of an inch.
No matter the outcome, good on you for testing. I bet you're one of the very few here in literally years that has bothered.
Paul