Using a shotgun and hypercardioid at the same time?
 

Does anyone do this?

For run and gun shooting, you could figure out a way to mount both a shotgun and hypercardioid onto your camera. In indoors situations, you won't get the weirdness/hollowness from the shotgun picking up reverb off-axis as in post you could flip to the hypercardioid. Outdoors, you have the shotgun which is more directional.

The downsides I can think of are:
You can't use another microphone. If you want an interview mic, you could take the hypercardioid off and use that (watching out for handling noise and making sure to keep it on-axis). If that doesn't work, it shouldn't be a big deal to switch another mic in. This isn't really a downside.

Weight. The additional microphone would add a little weight.

Price.

Shortened battery life (minimal).

Well...
 

A shotgun is technically a "hyper-hyper-hyper-cardioid" mic that happens to have very good off-axis rejection qualities.

Adding a hypercardiod mic in the hopes that it will give you good dialog pickup while avoiding the hollow room tone effect probably won't really give you much.

The hypercardiod will probably still pick up a great deal of room tone (more than the shotgun would) which may well be worse sounding than the hollowness you're currently hearing.

Your mileage may vary, of course, so the best route would be to run a test: do a mock interview with a Shotgun then swap mics to the Hypercardioid and run the interview a second time and compare the two on good speakers. Which do YOU like?

Well supposing you want to pickup music or PA sound (indoors), wouldn't a hypercardioid sound a lot better?

As far as dialogue goes, you probably want to get right up to someone. With a shotgun, the hypercardioid might have the added advantage that you don't have to point it right on. If you get really close to the subject (which you want to do anyways), a hypercardioid might give you better framing options?

Clarification!
 

Ah, yes: most definitely. Sorry, I thought you were looking to keep the sound focused on a specific spot, ala Dialogue.

Okay: the less off-axis rejection you're working with (cardioid, hyper-cardioid, and linear (shotgun) are all examples of increasing off-axis rejection) the less room tone you're going to pick up. If room tone is part of what you want, then by all means: back off towards the carioid or even omni pattern mic realm. Part of the reason for the "hollow" sound of a shotgun is that you're getting reflection noise... but only from one direction. The "natural" reflection pattern of the room is being rejected by the mic and that leads to a very odd sound. Dampening the room may be your only option here: heavy curtains are best for short-term fix here.

No mic captures sound better far away than close to the subject but something like a parabolic. Shotguns and Cardioids (and Omnis for that matter) all capture the same LEVEL of sound energy: but have different amounts of rejection of off axis sound which gives you a finer control of what energy you're going to capture.

So yes, a Carioid will give you the same level of sound as a shotgun placed equal distances away (assuming a similar type of and sensitivity of capsule: a good condenser will out-perform most dynamics, for example), the difference will be in how much off-axis "ambient" noise you'll pick up.

Best way is still to experiment! Past about 10 inches or so a carioid/hc mic will start to pick up equal amounts of room tone to speaker tone which will start to lose the speaker in the mix. YMMV!