I've shot 4 five shot groups at 100 and I'm getting about twice as much horizontal as vertical stringing. Before these groups, the scope shifted in the rings so I remounted the scope tightening the rings. Not farmer tight but definitely more than just using the screw driver with my fingers. A little bit of forearm strength this time. Scope has the parallax adjusted to 100 yards. Barrel has been floated and the action bedded. Any help is appreciated.

Handloads or factory ammo? What are you shooting?

Handloads. 308 win with 155 Scenars, 45.5 Varget, 0.030" off

I did shoot two other rifles on the same day, and I didn't have the same problem. So, I don't think it's technique.

Based on some experience, I would suspect the scope or the mounts/mounting.

Proven scope? What kind of mounts and rings? Bases good and tight and loctited down?

It's a new Bushnell LRHS so I'm doubtful it's the scope. Maybe it's the mounts. I'm using Talley LW mounts. Bases are tight for sure. Rings are a little tighter than I usually go. Maybe they are too tight??? I'll mount a proven scope with different mounts. If the scope is canted in relation to the bore, would this cause horizontal stringing??

The groups are 5 shots and go about 1.5" horizontal and 1/2"-3/4" vertical.

That is a very accurate bullet in a rifle that shoots well. Does this gun do the same with a different bullet? I`d start there.
Also, unless the gun is proven to shoot well, because it`s bedded and floated, dose not guarantee accuracy. I`ve seen some wicked "bedding" jobs.
I would also ask how the bullets fall...1st shot 2nd shot etc. My experience suggests that a three shot group of two together and one out, will go two together, but at different POI, with four shots. I would suggest bedding the problem.

Parallax adjusted by visual inspection or by the # read off the knob?

What action and model designation?

I had a load for my 270 using a stiff charge of RE22 under a 130gr TTSX. I got just over 3,200fps and was stoked to use it, but had horizontal stringing.

I tried different brands of brass, primers and seating depth, R & R'd the scope and bases, even rebedding the rifle, all to no effect.

What cured it in my rifle was H4831SC and IMR 4350, though I did lose about 50fps.

Did you lap the scope rings? Sounds like you may have bent the scope tube when you tightened it down the second time.

Before the scope shifted and you tightened it down, was the rifle grouping ok?