Here's a current photo

If the hoop-house is up and as warm as the starting area, or nearly so, then you can put them in it now. If instead the hoop-house is not getting - and staying - hot enough yet then I guess you will have to wait. Once you put them in it you want it to be warm enough for them all the time, including at night.

A friend that does this goes through a phase where the plastic is being pulled back part of the time, then I think grows them during the hottest time of the year with little or no covering. I've lost tomato plants to late blight(?) in August(!) so if I was going to grow them again I'd probably try to have them under cover most of the time.

Nice-looking seedlings.

Transplanting directly into 5-gallon buckets would eliminate one transplant-shock slow growth week, but exposing plants that small, even though "Sunblasted", to a very sunny day may burn the leaves. Perhaps you have shade cloth to drape over the hoop house.

So they need to be up-potted soon as the roots are likely already near the inner surface of those small pots.

And of course they may outgrow the lighting system / tray spacing setup, forcing the move to outdoors.

Tomatoes that young would prefer temperatures around 80 and not going below 60 at night.

If you have grown tomato starts in your hoop house in previous years, you already know that one sunny May day can cook them in an hour unless the plastic is pulled back, as mentioned above.

Not sure what the best 5-gallon bucket medium is for this. Go easy on the peat moss or the mix will be too acidic.