Switched outlet needs to be on for an exterior light switch to work; how can I fix this?
You have the wrong wire identified as your always hot
The problem you have is that the last installer who was in the box made the same mistake and identified the incorrect wire as the always-hot for this circuit.  In particular, based on what happened when you moved the jumper to the other terminal on the switch for the receptacle, the wire pointed to by the red arrow in your pictures isn't the always-hot coming into the box, but the switched-hot going off to the receptacle. In turn, this makes the wire pointed to by the green arrow in your pictures the always-hot coming into the box.
So, you can fix this by undoing the wire nut on your pigtailing job, removing the wire from the top screw of the switch for the receptacle, nutting that wire in with your pigtails, and landing the wire you undid from the pigtailing job on the now-free top screw of the switch for the receptacle.  Once that's done, you can button the box up, turn the breaker on, and enjoy your lightswitches!