Completely new to woodstoves...how to engage (and what is) the after burner on a TL300?

Yup, when you close the damper that forces the gasses down and into the rear secondary burn chamber, the AB. Sounds like you are getting the stove hot enough, but the other keys to making sure the secondary burn stays going is to 1) use wood that is good and dry and 2) have a deep enough coal bed established. (I prefer wood cut/split/stacked for a full two years.)

When I ran a Harman the AB would sometimes not get going, or stall out, if these conditions were not met. In the case of the coal bed, it just means opening the damper back up for a while. In the case of wet wood, it means you're screwed. What happens then is you get a very inefficient and smoky burn and creosote buildup.

To make sure it is running right, just go outside and check your stack after closing the damper. You shouldn't see anything. No smoke = success.

My Harman was the cast iron Oakwood, with no heat shields, so I could easily measure temps anywhere on the stove with my IR gun. Because the secondary burn is in the rear AB (also called the "combustion package") the stove would throw a lot of heat from the back. Temps on the top could be 500-700, but somethimes the back would exceed 800f. If it was only 300-400, I knew that I wasn't getting a secondary burn. I'm not sure how to check the rear chamber temps on the TL steel stoves, though.

If for some reason you find you are getting a smoky burn even when doing everything right, it is possible the AB has gotten clogged or has degraded. This does not happen to all of these stoves, but is not uncommon after several years of hard use. Otherwise, it should be a real heat monster. Good luck and happy burning!

On a reload, you will usually still want to wait a few minutes before closing the damper again, just to get the load charred and hot, to be sure you don't stall out the AB. Especially so if you let the fire complete the burn cycle rather than reloading only part way through the cycle. But the time is a LOT shorter than from a cold start, where there is no coal bed.

I would generally open the air control all the way up when reloading, and then step it back in few stages after closing the damper.