I'm not an expert in this field, but I'll give this one a whack. Maybe someone with more experience in vibrations can tell me where I'm wrong.
It's possible that the new muffler has a relatively thin case. If that is true, then it can resonate as the exhaust pulses cause pressure changes in it. This can manifest itself as an annoying buzz or hum inside the passenger compartment, especially if the sheetmetal above it resonates as a result. I've had this happen to me with cheap turbo mufflers.
I suppose the muffler angle could conceivably have something to do with this, as the angle of attack of any resonance or sound waves is now more perpendicular to the floor. I'm not well-versed in NVH (noise, vibration, harshness) so I'm kind of out of my depth on that aspect. The heat shield may have also attenuated some of the vibrations.
First, make sure the muffler shop didn't forget to reinstall any of the rubber exhaust hangers. Also make sure that when they installed everything, they didn't misroute a pipe. A pipe rubbing on some sheetmetal can make you go crazy after a few minutes.
If you're bored (or desperate), you can temporarily add some mass to the outside of the muffler. Strap a small (non-combustible!) weight onto the top of the muffler with some hose clamps (a 5-lb barbell weight might do it, though some lead sheeting, if you can find it, would be better) and see if the resonance diminishes. Even a few hose clamps wrapped around the muffler might do it if they can keep the sides from flexing. If the resonance diminishes, I would guess that the relatively thin muffler case is at least part of the problem (ever notice how thick the case is on, say, a Hooker Aerochamber?)
If you can replace the heat shield, I would do that as well. I don't think the tailpipe has much to do with the resonance unless it exits underneath the body instead of at or just past the bumper. I found out about this one the hard way just two days ago. The tailpipe broke on my 95 9C1, just behind the muffler. I limped the car home and sawed both pipes off just after the mufflers (Aerochambers). The exhaust now dumps in front of the rear axle. I knew it would be louder, but I cannot believe how much drone there is now. The resonance is enough to split your head. All those sound waves are reverberating against the floorpan just beneath the rear seat. I was going to drive it as-is, but I may have to bite the bullet and get tailpipes.