Taking a lead from Dej Marís posting, and assuming that the blindfold will not prevent complicated manoeuvres, I would follow a similar path, but make the first straight section longer than 1 mile before joining the unit circle. At first sight this would seem to add to the journey, but it actually allows the circular arc section to be shorter if the length is optimised. Here are the steps, starting from O.
OPStraight line, more than a mile, in any direction from O. PQTangent at Q to the circle with radius 1 mile and centre at O. Let A denote the angle POQ. QRCircular arc subtending an angle 3*pi/2 - 2A at the centre, O. RSStraight line of length 1 mile, tangent to the circle at R.
Note that the arc length has been chosen so that PS will be a tangent to the circle, and therefore any line which meets the circle must also meet the path OPQRS. This is equivalent to saying that the path OPQRS must meet any line that is within 1 mile of O.
Path length = OP + PQ + arc QR + RS
L= sec A + tan A + (3*pi/2 - 2A) + 1(1)
dL/dA=sin A sec2 A + sec2 A - 2
For L to be a minimum length:sin A sec2 A + sec2 A - 2 = 0
which givessin A + 1 - 2 cos2 A= 0
sin A + 1 - 2(1 - sin2 A) = 0
2 sin2 A + sin A - 1 = 0
(2 sin A - 1)(sin A + 1) = 0
sin A = 1/2 gives the only acute angled solution: A = pi/6, and substituting in (1) then gives the minimum length as