I think we're going to need a bigger box
- Five Romex cables entering the box (presumably all 12/2) = 10 wires
- one more Romex or armored cable you plan to add = 2 wires
- the transformer = 2 wires
- all grounds together are counted as 1 wire
- all cable clamps together are counted as 1 wire
total 16 "wires". At 2.25 cubic inches per wire, you need 36 cubic inches. The 4" octagon box you have is 15.5 cubic inches. If you get a box extension, that'll be 31 cubes, still not enough even if it was all #14 wire!
You also have a cable entering over top of the cable clamp instead of properly. On the right side you have 2 cables under 1 clamp. And you need to add a 6th cable.
So I recommend you stop by a real electrical supply house and acquire a 4-11/16"(120mm) square box that's 2-1/8" deep. That's 42 cubic inches. Then get a box lid that provides the lamp flange (+~6 c.i., giving enough for 2 more cables). Get six cable clamps that go in knockout holes (unless they are rated for 2 cables per clamp) and an armored cable clamp for knockouts. Don't go to big-box, they overcharge for these boxes and lids.
Before you take any of that apart, see the cluster of 3 black wires including the one to the bulb? Tape all 3 with red tape, as they are switched-hot coming from a) the switch loop, b) to this lamp and c) onward to another lamp.
Now look at the always-hot bundle, the one with 2-3 black wires and a white. That white wire needs to be marked with black tape; it goes to the switch loop. All the other whites (and one black transformer lead) are neutral. Adding one more neutral may be an adventure because there are already 6 under that one red wirenut. You need to split that to "3 and 4" with a pigtail between them.
That stairway light needs 2 switches (or no switches)
Code requires a light switch at the top and bottom of the stairway, so people aren't stumbling up/down the stairs to reach a switch. That necessitates 3-way switches, a fairly ambitious wiring job. The alternative is to hardwire the light "on" 24x7; at that point you need no switches at all. This now makes sense because of the crazy efficiency and long life of LED - a typical LED replacement bulb may be 5 watts, which will cost $5/year to burn.
You can also get bulbless LED fixtures, in fact new building efficiency codes require them in some places. (To be more precise, they require either a) any light sockets aren't Edison and won't fit incandescents or b) motion sensor... but motion sensors are problematic on stairs).
Wiremold surface conduit
One option for getting to difficult locations without making your house look like a Borg cube (looking at you, MC cable) is Legrand Wiremold surface conduit. This starts at any junction box and tacks neatly to the wall surface. You run single wires (THHN) in it, though you can cram in a cable if you really want to.