It's true you can't hear it when recording outdoors above ambient level, but it really does show up when recording indoors.
Being indoors or out doors has no bearing on your problem when it comes to "ambient levels".
{In atmospheric sounding and noise pollution,
ambient noise
level (sometimes called background noise
level, reference sound
level, or room noise
level) is the background sound pressure
level at a given location, normally specified as a reference
level to study a new intrusive sound source}
Anyway it's crap - it shouldn't do it!
EVERY electronic device will create its own noise floor! From micro motors to digital recorders, regardless of how much money you spend. If it has an electric power supply than it has a noise floor. It is mathematically impossible not to do so. If you have tried every suggestion in this thread and you still are not happy with the results, than you will at least have gained a little knowledge for your next purchase.
Honestly, Zoom makes different models of the H1. Which one do you own? You must also be aware that the H1 is absolutely the "Bottom of the Rung" digital recorders. Here is a video that may help you.
Also, are you mounting this on a tripod, or your video recorder? If you are using it via hand held, that will also effect your audio. What mic are you trying to use? Is the mic cable balanced or UN-balanced? Are you going from a 1/4 inch mic cable into a 1/8 inch reducer? Is your mic cable going from mono to a stereo reducer? At this point, I really have run out of ideas to help you out. Like I said, if all else fails, contact Zoom directly. I am quite sure they have spent millions on R&D for this product as well as their other recorders.