I'm trying to install a new outlet about 3 feet above an existing outlet (for a wall-mounted TV). It appears that the existing outlet is in the middle of the run, so I'm not sure how I would add the existing outlet. Can I connect the new outlet to what I believe is the plastic push connector? (There appears to be room for one more set of wires.) Or can I wire the new outlet directly to the existing outlet? I've included a photo of the existing outlet below.
Electrically, you can go either way
Attaching to the push connector or attaching to the backstabs or screws is 3 of one, half dozen of the other.
I mean, backstabs (and push connectors) are known to be unreliable, so you take your chances of having an hours-long, frustrating bug hunt. Screws are more reliable, but you must torque them adequately.
A better option might be to switch to a screw-and-clamp type receptacle, which can support 4 wires in special grooves under its screws (2 per screw). Assuming Code allows this; the choice of push connector for 2 wires is an odd one since there are already 2 receptacle backstabs. I suspect there may be a local amendment that requires the push connectors and pigtailing.
Cablewise is a different problem
This appears to be modern construction using steel sheathed AC cable instead of non-metallic. Note how there are no grounds in the box; those are on the AC cable and secured by the AC cable's clamps.
Nobody uses AC cable instead of Romex for no reason. This is certainly due to a local amendment requiring it. This means you also need to use AC cable or proper EMT conduit.
The problem is your cable needs to go upward. The two upward holes in the junction box are already occupied. This means you will have to come into the bottom and go upward, or relocate one of the existing cables to side or bottom if possible.
Yes you could connect to the connector or the receptacle, but I wouldn't go to the receptacle when an opening is available on the connector. Actually I never spliced through a receptacle in a commercial project (which I assume this is because of the data jack).
For me it would be pretty hard to fish Armor Clad cable into the box and properly ground in the self-grounding cable connectors without access to the outside of the box. I would look to find a 1/2" knock-out and use MC cable and pigtail the ground from the MC cable to the box and receptacle.
It looks like the receptacle has clamp connections (not stab-backs based on no release tabs on the receptacle), if they are clamps they are just fine to use. It also looks like stranded wire, do not wrap stranded around any screws, you would have to use stake-ons, the clamps backs are better.
Also check breaker size, if the wire is #12 like the pigtail wires it may be a 20A breaker, and #12 is the minimum size in most jurisdictions.
It definitely looks like there's an extra slot on those connectors so you could use those. You could connect your new cable to the extra screws on the outlet if the existing connectors won't hold an additional wire. The existing cable looks like #12 BX so check codes if you're thinking of using Romex. Good luck
There are wiring kits for TV's that are essentially two hole grommets and an in-wall extension cord that run up to the TV. They require no "wiring" because the TV just plugs into the existing lower outlet. I would love to have a high mounted outlet, but it's something to consider if it looks like the wiring might be too hard. If your code requires conduit, it's going to mean quite a bit more work than just fishing a few feet of romex through the wall.