Best not to consider the gears as "increasing towing capacity" just as making it easier for the truck to do it. I saw dramatic drops in towing and hauling trans temps on many trucks with lower gear ratios.
As to towing with medium-big tires and a 6-inch lift, I'm with Meep.... doesn't make for a great tow rig. Coupla things you can do to equalize it a bit: go with a rim increase and a shorter profile tire. If you rims are 16s or 17s now, go with a 17 or 18-20 inch rim with the metric equivalent of a 35 (305/70R-xx). The shorter profile tire will help with the towingby being a bit more stable but will still have the height. Also, investigate the LEAST amount of lift that will support that tire size on your truck. I suggest calling the lift kit mfr. and asking them about towing with the lift you are interested in. They may have some advice or options to make it work better.
As to towing with medium-big tires and a 6-inch lift, I'm with Meep.... doesn't make for a great tow rig. Coupla things you can do to equalize it a bit: go with a rim increase and a shorter profile tire. If you rims are 16s or 17s now, go with a 17 or 18-20 inch rim with the metric equivalent of a 35 (305/70R-xx). The shorter profile tire will help with the towingby being a bit more stable but will still have the height. Also, investigate the LEAST amount of lift that will support that tire size on your truck. I suggest calling the lift kit mfr. and asking them about towing with the lift you are interested in. They may have some advice or options to make it work better.