Agree with Palomine, chain link is a disaster waiting to happen when it's used as horse fence.
I would strongly suggest you get a good quality solar fence charger (if you want something that packs quite a wallop instead of just a little shock, look into chargers designed for cattle) and run some electric wire/rope/tape along the inside of your other, more solid fence. Most horses only need a jolt or 2 and they'll stop trying to get out.
Edited to add: If electric doesn't work and you can't find any other type of normal horse fence that will keep them in (no-climb wire, wood, etc) then the only option might be pipe panels that are simply too tall for the horse to try to climb and too sturdy for them to rub on and break down. The panels that I get are heavy duty pipe that won't bend, unlike those flimsy tubing panels that you get from the local feed store. I have 2 full drafts and 2 half drafts and it takes about a week for tubing panels to look like this at my house.
A good electric fence will keep in 99.9% of horses, problem is many people don't build a good electric fence. That .1% that wont stay in good electric fences need you to spend a lot of money on a very solid fence or sold.
Many horses have learned to go through electric fences just because it was never hot enough to teach proper respect. Most can rehab with a darn good fence, some you can't. But then some learn to shove another horse through the fence first so they can get out. I grew up on a farm and dad had one rule when it came to animals, if they refuse to stay in a fence get rid of them.
Does and Don'ts of good electric fence:
-Stay away from fence chargers that only are good for a couple miles, they are weak and tend to not last long. You want, at a minimum, a 20-25 mile fence charger. They are strong enough that animals respect them and last a good long time.
-Unless you have no outlet available, stay away from battery and solar chargers. They just are not as strong over time and definitely don't handle weeds as well.
-Stay away from tape. Yeah it's easy to see but those fine strands just don't carry the voltage and then when you tie a not in a break you lose even more. They also rot in the sun so after a couple years they break when it's windy. Use solid wire instead, then tie some streamers to it. Streamers let them see where the fence is when the fence is new. Once they know where a fence is they are no longer needed.
-You have to have a good ground, without one you are wasting your time.
-Run 3 wires. Top and bottom are hot with middle tied to ground, that gives them a very rude wake up call when touched.
-Keep fence grass and weed free. Walk your fence at least once a week to inspect it and remove vegetation. I normally spray roundup all the way around.
-Get yourself a fence tester and use it.
-Check your insulators. Spiders, dirt and old age can all allow your wire to short across to the pole.
-Once you have a good strong fence, get your horse shocked by it. Have to be careful doing this as you don't want them to jump on you but do it. You want instant respect for that fence.
-Maintenance, maintenance, maintenance. They'll try the fence along just to make sure it's still hot. If it's not they'll go right through it, this is why you don't want a charger that weakens over time.
I have used ALOT of different fencing. For my money there is just no beating electric fencing. It will even stop jumper many times. A good weed cutter charger, and set it for "extra crispy" LOL takes a couple whacks and the horses learn. I only have mine on a small redhot 15 mile charger now because the horses just don't push their luck. Hotwire is cheaper and more effective than any of the rest.. especially if you have cribbers.. they don't seem to chew on hot wire.. I have mine set up where it is a continuous circuit. All my gates have an up and over wire (on 10 foot t posts) so that when I open a gate it is uncharged but the rest of the fence stays hot. Any way you run it though make sure that your gate discharges when you open it, never lay a hot gate on the ground unless you somehow like grass fires...