I don't think anybody is making a DA six shooter .22 currently, so you'll be hunting the used market.
Smith and Wesson is still marketing the Model 17 Masterpiece, which is a classic, blued six shot double action 22 revolver. But it is pricey.
https://www.smith-wesson.com/product/k-frame-150477
I do share your feeling of not wanting an eight shooter or a ten shooter. I prefer six shooters. More holes in the cylinder just means you go through more ammo faster.
As an unabashed Smith and Wesson collector, I feel they make the very best 22 caliber revolvers. I have some Rugers, but I prefer the high quality of Smith and Wesson. I also have a couple of old double action Colt 22 Rimfires, and the quality is just as good as S&W, but they are relatively difficult to find, and tend to be expensive.
Allow me to take you through some history of 22 Rimfire S&W revolvers.
The very first revolvers Smith and Wesson made in 1857, if memory serves, were the tiny #1 Tip Ups. They were chambered for what we would call today 22 Shorts. Here are three #1 Tip Ups. At the top is a #1 1st Issue, Fifth Type which left the factory in 1859. In the middle is a #1 2nd Issue, sorry I do not know the date it shipped. At the bottom is a #1 3rd Issue which shipped in 1870.
A word about Tip Ups. This is actually a 32 caliber Tip Up, but it shows how they worked. A latch at the bottom of the frame is disengaged, and the barrel rotates, or tips, up. Then the cylinder is removed. Spent brass is poked out of the cylinder with the rod mounted under the barrel, fresh ammo is popped into the chambers, then the cylinder is popped back onto the frame and the barrel rotated back down and latched, ready to fire. They were very simple revolvers, and were never made in any calibers larger than 32 Rimfire, but they worked quite well.
Smith and Wesson never made any Top Break revolvers chambered for 22 Rimfire, The smallest Top Breaks were chambered for the 32 S&W round, like this little 32 Single Action from 1881.
Smith and Wesson began building revolvers with swing out cylinders in 1896. In 1902 they made their first 22 Rimfire revolvers with a swing out cylinder, the tiny M frame Ladysmiths. The Ladysmiths were seven shot revolvers chambered for 22 Long, they could not chamber 22 Long Rifle.
Three different models were made, from 1902 until 1921. Left to right in this photo the Ladysmiths are 1st Model, 2nd Model, and 3rd Model. There were a few Ladysmiths made in a target configuration with 6" barrels, but the gun was too small to really be useful as a target revolver. There is an interesting story that when staid old Daniel Wesson heard that the Ladysmith was the preferred firearm of ladies of the night, he ordered production to cease. Not true, Wesson died in 1906, the Ladysmiths were in production until 1921.
In 1911 San Francisco dealer Phil Bekeart thought S&W should build a suitable 22 Rimfire target revolver. He proposed a 22 caliber revolver be built on the I frame, which was a bit bigger than the tiny M frame Ladysmiths. Smith and Wesson made up 292 of these on special order and shipped them to Bekeart. Known as the 22/32 Hand Ejectors, or 22/32 Heavy Frame Targets, because they were bigger than the tiny Ladysmiths. This one is a relatively late one, it shipped in 1940. Notice the tiny rear sight that was adjustable for windage and elevation. The oversized 'Bekeart' style grip was needed because the frame of the revolver was fairly small, and a large grip was needed for a target revolver.
In 1931 Smith and Wesson introduced the K-22 Outdoorsman. This model was built on the K frame, hence the name. The K frame was designed as the perfect size for a 38 caliber revolver, but a 22 Rimfire revolver could be built on the same frame, the caliber would simply be smaller. In my opinion, the K frame 22s are the finest 22 Rimfire revolvers ever made. Big enough for target shooting, but small enough to be comfortable for woods carry, or simple plinking.
This K-22 Outdoorsman left the factory in 1932 and shipped to the Providence Rhode Island Board of Public Safety. It was probably used by the Providence pistol team. Don't let it's worn appearance fool you, this one is the most accurate 22 Rimfire revolver I own. Because of its beat up appearance I was able to grab it for $600 a few years ago. Notice the tiny rear sight which is adjustable for windage and elevation. This revolver is wearing incorrect Magna grips.
Another K-22 Outdoorsman. This one shipped in 1935.This one is in much better condition, and has the correct service grips.
This is a K-22 Masterpiece. The K-22 Masterpiece was issued briefly in 1940-1941, but production stopped for WWII. Production resumed in 1946 at the end of the War. This one is a Post-War K-22 Masterpiece, it shipped in 1950. Notice the large screw for windage adjustment on the rear sight, and unlike the Outdoorsman, the Masterpiece had a rib running down the barrel.
This is a K-22 Combat Masterpiece. The 4 inch barrel is standard on this model, and the front sight is a Baughman Quick Draw sight. With its short 4" barrel this model is very desirable for walking through the woods. This one shipped in 1953 and I stole it a few years ago for $800.
In 1957 Smith and Wesson changed over to a Model Naming system. The K-22 became the Model 17. I bought this Model 17-3 brand-spanky new in 1975. It cost $125, which was a lot of money back then. Notice the barrel is now a bull barrel without the slight taper of the previous two models.
In this photo, my Model 17-3 is at the upper left, two K-22 Outdoorsmen are on the right, and the little I frame 22/32 Bekeart is on the lower left. Notice how much smaller the Bekeart is compared to the three K frames.
Anyway, that is a short primer on Smith and Wesson 22 Rimfire revolvers. If I was looking for my first double action 22 Rimfire revolver today, I would be looking for a good used K fame Smith. If they are well cared for they will last forever.