When I've used the H4n with an external mic, I find that an input setting of about 35-40 is best (since anything beyond this the noise get's exponentially louder and the S/N exponentially smaller. For loud things like a Union Pac freight train by close up, I'll record more at 20-25 input level. That seems to still give enough hedroom to properly limit/compress in post without too much of a noise issue. The problem is if you turn the gain too high, you'll start clipping or hitting an internal limiter, which will mean that at some point you will actually be reducing your S/N. The 35-40 area seems to be the sweet spot in my experience where you get the most bang for buck in terms of S/N and signal robustness.
But it could also be a P48 issue. As I recall the 416's only power source is P48 (unlike an NT4 I own which can also run an internal 9v battery for its own power source). I've noticed the P48 to be weak on the H4n in my experience, and it drains the battery really fast when I've tried it. So in that case, even though maybe you're feeding P48 to the 416 and it's actually 'working', it may be underpowered, which could essentially lower your S/N and sensitivity range compared to using a robust P48 source like a Sound Devices or FR2 for example, even though the mic still powers on.
My personal thought is to adjust your signal chain during recording to maximize your dynamic range while finding a nice balance for minimal noise (for a signal chain item like the H4n it's more straightforward than say, a chain of 3 gain stages from mic to recording medium) - and then in post do the rest since you have non-destructive control. Whereas out in the field, if you screw up the signal by recording way too low or way too hot, it's already toast. A DAW will introduce such an acutely minimal amount of noisefloor that adjusting the gain in post, especially if you recorded in 24-bit, won't be an issue. That's my personal suggestion.
Noise is relative too. If you master a Foley sound really hot for instance, sure, there may be some noise even on the best of preamp. But if you know the editors always going to drop that sound by something like 20dB when they use it in their edit, that noisefloor all but disappears. not that you don't want to worry about noise, but also consider the relative context of the noise issue to the overall sound in question.