There's basically three kinds of dirtbox: distortion, overdrive, fuzz. Overdrive can be clean (a booster or line driver) or dirty. An OD gets some surprising tonal increases out of just about any amp, tube or solid-state (not so much with some sims, though). I have a simple homebrew OD (741 op amp) that's pretty clean up to about 7 before some nice clipping kicks in.
A preamp at the guitar end of the circuit is a line driver. Done properly, this makes the signal hot enough to push down the noise floor even as it preserves tone for long cable runs.
The TS is... well, it's okay. That is to say, most of the dozen-or-so circuit variations that have appeared as "Tube Screamer" are pretty good. But I used to have a gray-case DOD 250 Overdrive Preamp that was my all-time fave; I've since heard that the MXR Distortion+ & the Ross Distortion are mostly the same circuit.
Anyway, yeah, it makes sense to use an OD to push the amp, & a distortion to shape the tone going into it. You get, so to speak, BOTH burn & crunch.
As for hiss, that depends more on the circuit quality. There's so many players who use 10-20 chained pedals that I'm often surprised there's ANY non-hiss signal going to the amp.