I was much like you and was lucky enough to pick up a Rough Rider whilst leveling my Zane.
This methodology will vary on play style but I actually worked on the undercover skill tree first because in my mind, a dead character contributes 0 DPS. The early skills aren’t flashy but they do provide survivability to enable CCC in the mid to late game.
However, if you have no issues with survivability then I might suggest working down the hit man tree until you reach Death Follows Close. This one-point game changer makes ALL kill skills stronger and last longer. This includes kill skills in other trees coughsDonnybrookcough
If you are leveling and want to take a tier 1 skill in Doubled Agent, I might suggest Synchronicity. Since you’ll have Barrier and Drone, getting a damage boost for BOTH action skills active is quite nice and plays into CCC as it rewards more damage for action skill uptime.
Hope this helps, I’ve left it vague as Zane can build in a myriad of ways. Try different methods and see which ones work for you