I need to run a new wall outlet directly below a 4 switch wall switch. One switch controls an adjacent outlet, and I'd like that switch to also control the new outlet. Currently, that switch has a black and red wire. How do I wire this?

Can you use the other outlet and wire from there to new outlet and both is on that switch and simplify it. You say black and red wire. Does this switch have a sister switch located in another spot and both of these switches work together? If so it's a three way switch and not a single pole switch. If it's a single pole switch and just has Red Black and ground on it and the neutrals are tied back into the box together. The one wire black or red will be the feed to energize the switch and the other wire will go to what ever you are turning off or on. If that's determined you can run the new outlet off the same wire that turns the other outlet on. If its still confusing have switch off and use a meter and see what wire is always on or use a cheap test light to get the same results. The juice that on with the light switch of is the feed th other is to the outlet. That's the one you need to energize the new outlet. Don't forget the grounds and the neutral has to be tied into that same box. Hope this helps and Happy Holidays