How to unswitch a junction box so it is always on?
To answer your specific question requires you remove the cover plate from the switch, take a photo of the wiring, and post it here.

If your wiring is unfavorable, consider a wireless switch.  Newer versions of these use the kinetic energy of flipping the switch to power the switch, so no batteries and no restrictions on placement.

You could also put the "switch controlled" light on a timer, so it goes off automatically after a few minutes.  This lets you use the switch as originally desired, and also gives you a chance to control the motion light (many of which take a quick flip of the switch to mean 'stay on forever').



In general for anyone wiring an outdoor switch and light, strongly consider running 12/3 or 14/3 wire, even if the light is supposed to be switched.  This means running an extra "hot" wire for future use.