How can I extend the height of a wooden fence from?

If this were my fence I’d bolt some three-foot long extensions on the sides of the existing posts at the top. Overlap the new wood twelve to sixteen inches over the old posts and use two 1/2”-diameter hot-dipped galvanized carriage bolts per joint. If you install lattice on both sides of this extension it will hide the joint quite nicely. Today’s pressure-treated lumber is more corrosive than the stuff sold previously, so it’s essential you stay away from electro-plated nails, screws, bolts and hardware. Even contractors don’t always understand the importance of choosing outdoor fasteners well, so you need to be your own watchdog. Insist on hot dipped galvanized bolts. They’re a duller grey than the shinier, less costly and more widely available electro-plated alternative.