I just went through all this last fall and this spring. Or, I went through it with the help of everyone here, and Randall in particular.
My end float went down to zero after rebuilding and replacing the bearings. I ignored that at first, and thought I'd check the operation on a bench test using a hand drill motor. When the hand drill motor wouldn't spin the output shaft very well, I figured something was wrong, and others here did, too.
So I tore everything apart and discovered that I hadn't seated the head bearing far enough against the annulus. I'd made two mistakes. First, I was too gentle with my Harbor Freight press. Second, I had used a wood spacer for the pressing, and I was likely misled by the compression of the wood. With a steel spacer I was better able to hear and feel the bearing popping into its correct place. Admittedly, a better eyeballing would have helped as well.
OK, with the bearing better seated, and then reinstalled the rear bearing the rechecked the end float. This time it was .003, but only by pressing my thumbs with some real force into the annulus and watching the dial gauge.
After some debate, I decided that was good enough. That conclusion came from: a) Others here not expressing much horror; b) my suspicion that replacing my 50 year old washer would be bad luck and just more fixing of something not broke; c) my hope that the new bearings would adjust and give a little play after some break in; d) my guess that my measurements weren't so perfect anyway; e) my hunch that a British car shop back in the '60s would have likely slapped it together and not bothered with measurements anyway; and, finally, f) so what if I break this apart and have fun again one day!
It's been running great for 480 miles, so unless bad things come fast without notice, I think I'm good.
Let us know what happens on yours, but I think Randall is right when he suggests you check the seating of the bearings and go from there.