What I do in that situation is to find the safest opportunity to cross the traffic lanes then queue into the left-turn lane as any other vehicle, which means I take the lane and wait my turn like everyone else. I strongly dislike filtering past waiting traffic when I'm in a proper traffic lane, and I never do it since it's an unexpected and unsafe approach to cycling in traffic in my area. However, regional laws and customs differ with lane-splitting and filtering. After making the turn, I find the safest opportunity to move to the proper area of the roadway whether that be back to the right-most lane, or left lane if I'm making another left, or elsewhere depending on circumstances.

"Safest opportunity" depends on road-surface conditions, traffic, etc. and varies, sometimes it could mean a block away, sometimes right up near the intersection. Common sense will generally suffice to dictate the approach.

The path you took looks fine in that image (to me at least), but of course it depends on the real-world traffic situation at the time you were riding.