Originally I tried putting a UHF antenna on the highest piece of furniture in my bedroom. Results were pretty terrible, I could get only 4 stations reliably.
Then I moved the antenna outside onto a short mast, just a few feet above the roof line. This improved things dramatically and I could now get 9 stations. Better, but still not great.
When I replaced the old VHF antenna on the other side of the house with a much larger and more powerful VHF/UHF antenna, I stacked the old antenna on top of my UHF antenna, and inserted an additional 5 foot segment into the mast. I pointed the VHF antenna at Lincoln and the UHF antenna towards Council Bluffs. I could now pick up the 7 VHF stations in Lincoln, while the greater height and slightly altered angle allowed me to pick up and additional 6 UHF stations in Omaha and Iowa, giving me a total of 24 TV stations.
This setup was adequate, though KLKN on the far side of Lincoln just barely came in. When I built my new computer, I upgraded to a Hauppage dual tuner, which proved to be far less sensitive than my old ATI single tuner. This made KLKN almost unwatchable, which was a real bother to me because that station is the local RTV affiliate. So I searched around and ordered the biggest, most powerful VHF antenna I could find, which turned out to be a Winegard YA-1713 high-band VHF antenna. Since there are no low-band VHF stations (channels 2-6) in the area, a high-band antenna (channels 7-13) made a low more sense. It finally got (almost) warm enough for me to work outside yesterday. I also replaced the regular dual-shield coax leads from the antenna to the amplifier with quad-shield and I raised the VHF antenna a couple extra feet, which theoretically should improve the signal even further. I was also able to refine the direction I was aiming the antennas. The results aren't quite as dramatic as I'd hoped, but KLKN is coming in completely stable now, which is all I really wanted.
The only nearby stations I'm missing are KFXL, a UHF Fox affiliate in Lincoln, and KYNE, the PBS affiliate in Omaha. Trying to receive KFXL would require another UHF antenna aimed at Lincoln and a highly complicated series of filters and signal combiners. Since they have only a single subchannel and I already get KPTM, the Omaha Fox affiliate, it's totally not worth the effort and expense. I'm not sure why I can't receive KYNE; it might be because they have a fairly small tower (117m). They have a total of 3 subchannels, but since they carry pretty much the same programming as KUON, I'm not really missing anything.