Yhyjyk

I would like to dig some 6-12 inch diameter dry wells (any diameter in that range will work). The holes need to be at least 6 feet deep to reach the permeable layer of the soil. However, the equipment the local rental agency has will only dig 4-foot deep holes.

What is the right equipment to dig 6-foot holes?

The soil is clay with some sand. There are no rocks. It digs very easy.

On a side note... Not saying you haven't already, but make sure you call the utility locating company so you don't cut through a buried cable, water line, or worse... Buried electric. Can be deadly.
It's probably in the high price range for a small project, but there are machines called hydro vacs which dig holes with vacuum power and high pressure water. It's actually what I do for a living. Where about do you live?

Depends a lot on the ground. The advice to call for utility locating ahead of time is spot-on, no matter what you do to dig them.

In my ground, an auger is generally useless, since it will find a rock that it can't shift just about every time, so an excavator or backhoe with a bucket is the only tool (short of a well drilling rig that can drill through the rocks) that's going to work, and the well-drilling rig is far too expensive to set up for such piddly holes.

Well, actually, a pick and a shovel and a clamshell post-hole digger might work, if you want to go cheap and physical, and speed is not of the essence.

In nicer ground than mine, you'd dig some 2 ft deep large holes and then use your 4 ft auger, if cheap and physical and not very fast were your guiding principles, but you preferred renting the auger to a clamshell. In my ground (if not using a backhoe/excavator) I'd go prospecting with the clamshell and then dig a large hole to move the bigger rocks aside and provide room to go deeper with the clamshell.

Beware of climbing into a hole deeper than waist deep that could collapse around you, since that can kill you, which could be regarded as "not cheap."