Your horse is both lacking in fat AND muscle. He may benefit from very gentle, slow exercise. Exercise for out-of-shape horses should start out with very brief, slow sessions. It has to be adjusted to whatever health problems or lameness/foot problems the horse may have, and exercise for older horses requires a lot of care and thought. It's not usually exciting or fun and shows and long trail rides can't usually be the goal - it's often boring repetitive stuff like hand walking, walking under saddle (once the horse's back isn't so thin!), and it may only be for five or ten minutes. Many people find it way too boring and repetitive, but daily very brief sessions are best.
Make all feed changes gradually over several weeks. I would not assume that the previous owners fed this horse the feeds you gave them. It's good to start supplements and bagged feeds gradually, and not assume their hay or pasture is as lush and rich as yours.
The main thing with feeding the thin horse is using sound basic equine nutrition principles. Horses need plenty of good quality fiber in their diet (largely provided by quality hay and pasture). Hay and or pasture should make up most of their diet by weight. The average horse should usually eat upwards of 20 lbs of hay a day, or equivalent pasture(not to suggest that a horse can immediately start grazing six or eight hours a day - it may be more like 5 minutes a day and gradually, slowly working up to longer over a period of weeks).
Your horse is not starvation-level thin, but since many here rescue horses, I'll mention that vets advise extreme caution at putting starvation level horses on rich feed and trying for rapid weight gain. They must be put on a very careful gradual program or severe metabolic damage can result.
Recently, it's been found that many horses tolerate fatty foods well. Many owners have added some sort of oil or fatty product to their horse's diet. Many concentrates (bagged feed, grains, etc) are now sold with more fat included.
There are many myths about protein in the equine diet. Many owners feed rather a lot of protein. Large amounts of supplements or high protein bagged feeds may be fed by some owners. High protein diets aren't always good for older horses. Such diets can strain already aging kidneys aside from not doing what is claimed. It's a good idea to keep elder protein levels about 10-12%. Some will feed rescued animals more protein for a little while, feeling it helps rebuild muscle and helps in healing....and then cut back to recommended levels.
There are also myths about vitamins and minerals. Many horse feed products supply vitamins horses synthesize themselves. Many advertisements make outlandish and unproven claims for the vitamins, minerals and other products they feed, because horse feeds and supplements are completely unregulated. The growth in feeding of supplements and 'nutritionals' in the last 15 years or so has been jaw dropping. But many of the oral joint products marketed, can't even be absorbed by horses, and may vitamins and minerals and other items, simply don't do what is advertised. Stick to sound basic principles of nutrition. Iron, in particular, may be fed in such excess quantities("It builds them up!") that horses may actually get sick!
My vet commented the other day - 'I will be seeing many, many founders in the next few weeks'. Founder/laminitis are conditions that can be caused by sudden feed changes or excess feed(not the only possibility, but common enough).
My vet made that comment because of recent heavy rains in our region (and many others!) causing a 'flush' of pasture growth. Quickly growing grass is high in sugar and can cause laminitis in horses as well as other types of animals. Be VERY careful about putting your old horse on your pasture.
Many sellers will say their horse is 'on pasture' but it may be so grazed down that when the horses move to a new pasture, they become extremely ill. Be careful!
Beet pulp is one of the safer feeds due to its type of carbohydrates etc(again, other very high starch/sugar grains can cause laminitis in vulnerable horses).
When beet pulp is soaked it is safer (soaked for a long time, starting with boiling water) choking is less likely - shreds are less likely to cause choking than pellets. As discussed often here, a few horses will choke on it even if it is soaked.
Some horses try to eat rather quickly, so some horses are safer to feed beet pulp than others. Any finely cut up feed can be choked on. So it's important to get to know your horse - how he eats and what he does well on.
With a senior horse, the focus should always first be on - WHY is the horse thin.
Not enough feed? This isn't unusual. Older horses often receive less medical care and less feed.
Inappropriate feed? Old horses may need something that is a little easier to digest. For example, rolled oats may be better for an older horse than whole oats in the husk.
Poorly selected feed? Some feeds may be 'appropriate' and the right quantity, but that horse may simply not like it and not eat it. Horses have preferences too.
Pain? Horses in pain from lameness or injuries to the back, neck, hind quarters, often eat less. A common cause of pain is chronic laminitis/founder.
Disease? Some horses develop diseases that cause their gut to not work efficiently. There are many possible diseases that could do that.
Poor quality hay? Many horses get over 90% of their calories from their hay. If it's poor quality they will get thin.
Teeth? Teeth in poor condition can cause a horse to not get all the nutrition in his feed. Much of his feed may even be 'quidded' - dropped from his mouth in little half-chewed wads.
Worms? Older horses are often more vulnerable to worms.
Management? Older horses fed with other younger horses are often pushed away from their feed.
'Old age'? There is nothing about old age per se that includes being thin....except that some older horses are suffering from disease, pain or lameness that may interfere with weight maintenance.
You will hear a zillian 'absolutes' in horse feeding - it's a subject that brings up very, very strong opinions and every horse person always thinks he's right and everyone else is wrong. Almost every horse owner has a 'formula' they swear by, to fatten up a thin horse - a favorite product or feed supplement. You'll be told you have to feed that!
What is important is that the horse receive a balanced diet based on sound time-tested, basic, sound nutrition principles. A very rapid weight gain should never be a goal - weight gain that comes too quickly can cause, again, laminitis as well as other problems. A good many owners have a 'fat eye', in that they want horses to be overly fat, so aiming for those real round contours is not always the best. Confirming with a vet the horse's current 'condition score' and setting up a feeding plan with the vet, can help.
If you are stabling thin horses, you may want to prepare evidence that the horse is being seen by a vet and is on a sound nutrition program, as well as information about when you purchased the horse and pictures of his condition at that time. Neighbors and passers-by seem to be increasingly telephoning animal control agencies when they see thin horses. They don't always understand that thin horses didn't always get that way due to the current owner.