In a gas forge HT, a muffle is of great advantage. In an electric oven the size we use for knives, it makes less difference.
I experimented with making an 1/8" steel muffle for my K24 HT oven. It was three 4"X4"X23" steel sheets welded in a "U". It came up to within 1" of the top and 1" from the sides. The knife was placed on the steel floor of the "U". It was open on the top to allow circulation. I would pre-heat the oven and muffle for half an hour before placing the blade in. I could not detect any difference between the results with or without it.
I have used large commercial ovens that had built in muffle walls (called a muffle oven) where the elements are behind the refractory walls, and smaller jeweler's muffle ovens. Many of these do not reach stainless HT temps. These ovens are primarily used to burn or dry things to a powder, or for melting things like glass and some chemicals. I have never seen a commercial knife HT oven that had a built-in muffle.
What does matter a lot is allowing the oven to pre-heat and fully soak the refractory. The difference between a 10-minute warm-up and a 30-minute soak at-temperature was notable. Both will show the oven at the target temp, but the amount of refractory soaking is not the same. I think this is why many of the older HT regimes had a pre-heat step below the austenitization temp. This brought the blade and oven to an equal temperature before the final few hundred degrees needed to convert to austenite. For most steels, this step is not needed for the blade, but it is a good idea for the oven.
You want the oven to soak for a good 15 minutes at (or close to) the target temp. The rate your oven heats up may vary, so time it a few times to see how long a cold oven rises to the target temp. If it is around 10-15 minutes, then set the timer for 30 minutes. If it takes 25 minutes for your oven to heat up, set the pre-heat timer for 25+15 = 40 minutes .... etc.
I program my HT regimen with a first step at 5° below the target temp (e.g. 1440°F). I set the hold time for 30 minutes. Then the next step is the target temp for the steel I am hardening (e.g. - 1445°F). When the second step reaches the target temperature, I place the blade in the oven. It only takes a few minutes for the 5° rise. To avoid just sitting there watching the oven heat up, read the tip below.
TIP:
The newer oven controllers have a wi-fi dongle option that will alert your phone when the steps are done, but not everyone has that. The below is a 10$ hack to get pretty much the same result.
Get a cheap two-event kitchen countdown timer. They cost about $10 on Amazon. Set event one for 30 minutes and start the timer when the oven is turned on. This is your pre-heat time for the oven. Set event two for the austenitization hold time. Place the blade in the oven, and when the temperature reaches the target, start the second timer. Drop the timer in your shirt pocket or clip it on your belt. That way you can do other things and will get alerted by the BEEP when the oven is ready to put the blade in, and when the blade is ready to take out.
(Adjust these times to fit your oven heating rate as noted earlier.)
One good use of this time is to tidy up the shop, take out the trash, and sweep the floor. Those tasks can easily be stopped when the timer goes off.
The timer is really useful when doing stainless HT where the heat-up and hold times can total several hours. There have been many folks who forgot about the oven for hours when distracted by another project. Sometimes it was OK ... and other tomes it was a ruined blade.
The timer is also great for doing tempering. Many of us have left a blade in the oven for hours or even overnight accidentally. I have fallen asleep in my chair reading while a blade was in the kitchen oven, only to awake three hours later. The BEEP would have let me know when to take the blade out and go to bed.