Go 2/sqrt(3) away from the starting point, turn 120 degrees and head 1/sqrt(3) along a tangent to the unit circle, then traverse an arc of length 7*pi/6 along this circle, then head off on a tangent 1 mile.

This gives a minimum of sqrt(3) + 7*pi/6 + 1 = 6.397...

It remains to prove this is the optimal answer.