A low Swr is mainly an indication of a good match between your transmiter and antenna. A high SWR will not cause high noise. You can receive a lot of stations by just using a wire (24 inches or longer) stuck in the antenna port of the radio. DO NOT TRANMIT when doing this.
Sounds like you have a lot of locally generated noise. Using a AM broadcast portable battery operated radio tuned to a vacant frequency may help you locate some of the noise.
A battery operated scanner tuned to the AM (not FM!!) aircraft freqs. (139.200mhz or so) can also be used. It will work a lot better with 3 element Yagi antenna (easy to make) to pin-point the noise source.
Wall warts, (the little power pack that plugs into the wall), battery chargers, WIRELESS GARAGE DOOR OPENERS, flourescent lights, TV's, satellite receivers, light dimmers and AC powerlines can all generate noise.