How do I move a phone jack to a less visible location? I want to move it along the same wall, just approx. 3 ft. down and 1 ft. over.

I would remove existing phone plate and see which way the wires come from. And locate studs to see if you can go 1 foot over the direction you need, if existing jack is above 4 feet, their will likely be a Cat(side ways piece of wood between the studs)Try pushing a coat hanger or something to new location to see if an existing clear path exist.
Drill hole through cat if needed. I like to use the Gel filled "beanies" to make connections, match the colors. Try not to short the wires.
Also may be a good Idea to unplug your line from test connection outside to preven getting a slght jolt should someone call in. If Isolated and you touch 1 at a time, you will be OK, and could connect live.
If you were super lucky, wiring may run past new location, and if some slack present, you could pull all or what you need.