
If you are looking for an Ancient of a specific item, then, for most items, it is most efficient to reforge. In closing, some builds are better than others, but these steps are universally true in regards to pushing any character's power level."Most Efficient" method to farm for Ancient items depends on what you're looking for. You might make some friends along the way I did! Then do some high-level group GRs and rank your three legendary gems this way. Build a support Barb/Monk and lobby shop for high paragon players to play with.
Support characters are easy to build, need little optimization, and fun to play.
High-level legendary gems: Pushing your gems past your highest leveled character's threshold is possible by building a support character and doing high-level group GRs. Thirteen level 60 augments give you 3,900 main stat that is a shit ton of power you are adding to your character. Ancient items naturally roll higher, and you can't augment non-ancient items. That means you need ancient items to augment (duh), so farm ancient items. Augmenting Ancients: The easiest way to gain main stat is by augmenting ancient items. LON Bomb is a GR80 build with ~900 Paragon and a GR100 build with ~3,500 Paragon. When someone says this or that build is an 85+ build, ask what Paragon allows it to do that. Main stat, main stat, main stat: MAIN STAT IS WHAT EMPOWERS YOU TO DO HIGHER GREATER RIFTS! This is why Paragon is so important. Simply missing a CDR breakpoint, other vital breakpoints, or necessary pieces of gear can sabotage the best-laid plans. This means all your LON gear needs to be ancient, or if you are running a regular set, you need all the most optimized items for that set. Item optimization: You need a good set of gear with good rolls.
There is a surefire way to do higher GRs. If sets like that existed, you would know about them due to their widespread use, or they would be nerfed by Blizz to make room for the others sets. The answer is there are no builds that will let you out-right do high GRs. I see this question asked more often throughout the Diablo 3 community than any other: What do I need to do to reach high GRs, or what item sets can I use to do 80-90GRs.