Great question and answers.
I've found that some of the best tornado chase days are also the best structure days. That's not necessarily the days with the highest tornado probabilities or outbreak days, but the days that feature those slight risk, cyclical supercells that put down strings of tornadoes. They often have amazing structure, and I think that's probably due to the ample directional shear those days often have, along with the preferred storm mode: discrete classic.
What about some parameters that work against tornadoes but for structure? One of the best is a fairly stout cap, and mainly a cap that's filling back in after your supercell matures. Air forced upwards through a stable layer often has clouds with a smooth, striated look to them. You get beautifully sculpted supercells at dusk and after dark when the boundary layer cools off and the bottom part of the storm is in that cooler, stable air while the updraft remains strong. So a fair amount of CINH, but after your storm is established. You don't want to choke off the updraft completely before the storm can go up. Once you see those photogenic striations, the tornado show is usually over.
Another that's contrary to what we look for when chasing tornadoes and was also mentioned above is the temperature/dewpoint spread. A low spread often favors tornadoes, but also means low and hazy bases. A higher spread might mean it's too dry for tornadoes, but those high bases are often much easier to shoot from a distance and are higher contrast.
Another parameter I look for when after structure: steep lapse rates especially in the mid levels (greater than 8 or 9). Those often give you that crisp, explosive convection, whereas more meager lapse rates (7 or 6 or less) often result in weak, watery updrafts (even if you still have good CAPE values). You can get really steep midlevel lapse rates when there is a lot of cold air aloft or a deep trough moving through.
The big one is location though. Where do we often have steep lapse rates, fairly high T/Td spreads, the EML is a big player with stout capping, and skies free of haze. The high plains. Colorado, Wyoming, New Mexico often have gorgeous storms for all of these reasons. You might have setups in Colorado and Arkansas both with 3000 CAPE and 45 knots of deep layer shear, but with radically different quality of storm structure. Setups down in the southeast are often plagued by weak lapse rates, a lack of capping that results in the entire warm sector going up in clusters weaker updrafts, and low, hazy bases. Extreme shear can compensate for that sometimes though. There was some amazing structure coming out of MS and AL during the 2011 super outbreak when there was 100+ knots of deep layer shear and helicity values well over 500.
