As you mention existing leaf pairs that may throw off the reader. Your question is about the appropriate timing! Typically threading the branch is most effective with smaller buds and more flexible tips. This also enables a smaller hole for the thread graft. Combine this with the beginning of the growing season for best results. Note: you can remove larger buds and early leaf pairs from a section of the branch to ease the insertion when dealing with thicker trunks or branches. ( just not the growing tips)
For some species I prefer to wire the branch into place during the previous growing season as it extends. This makes for easier thread approach at the desired angle and also less disruption to the thread graft during the process. Bthe asically the branch is in place for the beginning of the next growing season, allowing for the insertion and the full growing season for the hole to close and the branch to settle in. This thread graft would be typically be performed in March prior to to bud swell. At least in my location.
Here is a picture to illustrate the approach suggested above.
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