Aloha Perry,
I see your relay is on top, so the relay operating circuit is on the right and the wires to the starter and coil are on the left. The big brown wire supplies +12V from the main battery cable to the center blade connector. That center connection also has a second blade for a small brown wire to connect to the fusebox. In my earlier photo, you can see it powers the purple, or always hot, circuit. I wonder if your horns/hazard lights, etc has power, as I don't see a small brown wire in there - can you widen the shot to include the fusebox?
Can you hear (or feel) the relay operate when you turn the key to start? The relay could be stuck on, but there are other possibilities. Where does the big red wire go - hopefully not to the coil. Although both the starter and coil circuits are powered when the relay closes, they are electrically separate, This is very important, because when the car is running, the coil is still powered by the ballast wire (or a home-made direct wire). This will supply power to whatever else is connected to the + side of the coil. If you're a cheapskate and adapt a spare horn relay as a starter relay, you'll be surprised at how hot the wires get, as the wire to the starter solenoid will still be powered. A ballast wire won't have enough voltage to keep the solenoid closed, just get real hot (ask me how I know). If the coil had full 12V it might keep the starter going.
I'd pull the relay out and give it an ops check on the bench. You can carefully un-bend the 4 tabs and pull the metal case off. I must have tossed my broken relay out, but there is a small spring that keeps the output contacts apart until the relay operates. If that fails, they probably won't open back up.
Jeff