chain link to picket fence
Maybe a Lumber Yard can route-out the centers of your 4x4s. This would meant that they'd have to cut them in half, length-wise, then route them out (half and half) to the proper diameter of your existing posts. Also, your 4x4 posts would be 1/8" less due to the saw blade, so the Yard needs to compensate for that. Then, you'd have to pilot-drill a few holes in the 'sandwich' in a few places, up along the length of the sandwhich, though which you'll use bolts to keep the sandwich together. I think you'd want the saw-lines to be on the front and rear of your fence line vs. the left and right because your bolts need to be parallel to your beams. You'll have to cap the tops if the Yard routes-out the entire length of the half-posts. If you have them only go up 4' (or a short more higher than the top of your existing posts), then you won't need post-caps.
Now, remember - If you have to ever drill THROUGH your new fence posts (after they're installed) there is a steel post in there. Hopefully, you won't. Good luck!
My best to ya and hope this helps.
Jay J -Moderator
PS: God Bless America!