How can I make an outlet controlled by switch always hot?
Let's take the better picture here



There's 3 areas in the picture


The Wire Nut - You have two black and a white. Whenever you see something like that it typically means the white is what's running to your switch. That means the other black wires are your incoming and outgoing hot
The black wire on the outlet - This is likely tied to the same cable as the white wire we just discussed, meaning this is your switched hot
The other white wires - These are your neutrals. It's not clear why you would have 3, however.


First off, let's get this basic outlet out of there. Do yourself a huge favor and go buy a new one. What you want is a "preferred" or "commercial grade" outlet (example) that has side plates. Expect to pay like $2-3 for this (you only need the one). While you're there, pick up some yellow wire nuts. It's not a bad idea to also have a contactless power tester and an outlet tester (buy the better one that also tests GFCIs). Now, let's rewire.


Take the wire nut apart and put one of your shiny new yellow wire nuts on the white wire, mark it with a black marker (just draw a band on the wire so it's obvious it's not a neutral), and shove it to the back of the box. Now, attach the two black wires to the brass screws on the left side (as you're looking at the back of the outlet)
Take the lone black wire from the outlet and stick a wire nut on it as well. Push it to the back of the box.
Attach the remaining whites to the other side (silver). These "side-stab" connectors support up to 4 wires, just make sure you get the screws tight


Obviously, remove and re-attach the ground wire, but that's straight forward.