This might be an overly simple answer…
How about hog panel on the back or front side of your wooden fence. It has a 4"x8" heavy wire mesh 4ga., it comes in 50’ high by 16’ long and is cost effective at about $25 per panel. It would have to be securely fastened to the main fence posts to prevent it from being kicked out or ripped out (Structural screws). You could attach it to the front or back depending on the aesthetics. On the front of the boards it our greatly strengthen the fence from boards from being kicked or ripped out for a pass through and on the back it would make it so that they would have to use bolt cutters to get through the second layer. For added insurance you could put a layer of grease on the top of the fence to deter people from trying to climb it or on the hog panel to deter tampering.
I am guessing that there is only an area of fence 16’-32’ that would have to be reinforced to deter the cut through. From what you described your problem is more of nuisance trespasser looking for a short cut. If you make it annoying enough they will stop messing with the fence and walk around.
At $100-$200 dollars it is worth a try.
below is a link to an example hog panel for sale:
https://www.tractorsupply.com/tsc/product/feedlot-panel-cattle-16-ft-l-x-50-in-h
Structural screws (I like this one due to the uncommon spider drive bit used to install and remove them)
https://www.amazon.com/FastenMaster-FMHLGM278-50-Heavy-Duty-Flathead-Fastener/dp/B001OEJULG/ref=pd_cp_60_3?pd_rd_w=HlkKQ&pf_rd_p=ef4dc990-a9ca-4945-ae0b-f8d549198ed6&pf_rd_r=FPHWES17FWQE828BHNNJ&pd_rd_r=3744db37-9eb1-11e9-acd2-e9c7b410e391&pd_rd_wg=nbmdw&pd_rd_i=B001OEJULG&psc=1&refRID=FPHWES17FWQE828BHNNJ