there is no way to get the thread to the other side without cutting it.
You can knit the bottom and drop the thread and cut it, knit over the top and pick up the reinforcement thread when you get to the bottom again and repeat. this will cause an enormous amount of sewing in. or.........
You could use 2 separate strands of reinforcement thread, start them on opposite sides and pick up the appropriate thread when you get to that side. less sewing in but it will be more obvious.
but if you dont want to go thro all that, you will have to knit it along with the sock yarn all the way around the sock. its not really all that obvious, and more "supportive" and sturdy. i use reinforcement thread in that same area too, but i take it all the way to the toe. Just add it at whatever point and finish the sock. its very neat, and feels great on the feet.
(my job requires alot of walking and doing my socks this way really makes a difference) Ive even done a pair of my daughters socks this way and she wears them everytime she has a track meet.