Hcfyk

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.

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."

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.

I dug some deck footers a couple years ago and my rental shop had 10" auger bits with a two-man auger (4 handles and a motor in the middle) for about $100 a day, made quick work of several holes like you're talking about (I dug 7). The bit itself was only about 3' long, but they'd also rent an extension bar if you wanted to go deeper (I'm sure there's some max depth but it went as deep as I wanted.

Some of the other options presented are not cheap (hiring a backhoe or tractor-mounted auger) or very physically straining (clamshell, hand auger) so depending on your budget, physical fitness and local availability of tools, these two-man motorized augers might be a good option.

Depends on your budget, and the value of your time.

1) Hand auger

These things are backbreaking - but if you have unlimited time they will eventually do the job. Notice the extensions? You can drill until you run out of extensions or the ground is too hard to cut. Performance is improved by using a sharp cutting edge, just like a sharp spade cuts better than a blunt one.

Note any rocks stop these things dead, so you have to use a crowbar to dislodge the rock, or dig another hole and hope to miss them all.

Also, a 4" / 100mm hole is half the work of a 6" / 150mm hole and is about 1/10th the work of a 12" / 300mm hole.

On the plus side, this tool is character-building, and great work for students on workdays :)

2) Tractor Mounted Auger / Post Hole Digger

This one is front-mounted

These are also found mounted on the rear PTO

For deeper holes, you use a longer auger

Naturally you would hire this tool rather than buy it. If you have no tractor then its better to just get a contractor to do it - probably drill all your holes in a morning session if you pre-clear access and clearly mark the locations before-hand.

My personal favorite is vacuum excavation. It's nothing fancy nor expensive; a respectably large wet-dry vac does the job nicely. By "respectably large" I'm suggesting something with a 2.5" suction hose and a 14 gallon or larger capacity.

The process is simple. Alternate between a tool that loosens the soil and the vacuum to remove soil from the growing hole. With two people working in a moderately sized hole the steps can be done concurrently. My tool of choice for loosening soil is a "digging bar." It's a straight steel bar about 6 feet long with a flat on one end and a point on the other. Occasionally a rock or clump of soil will block the hose and it'll have to be cleared, but progress is relatively quick and I find that it's much less effort than the traditional clamshell post hole digger.

Whereas your holes are to be 6 feet deep, and this is about the same length as the digging bar and also the pipe included with most wet-dry vacs, it may be necessary to extend these. Purely by way of example, Home Depot (US) offers an "extension wand accessory" for their Rigid brand vacuums; one or two of these may be enough to reach the bottom. The digging bar could be extended by welding or otherwise attaching a length of steel pipe at the end.

By the way: don't forget to empty the vacuum drum regularly. It turns out that 14+ gallons of soil is heavy!