You've misinterpreted the movement penalty for grappling and the movement restrictions for jumping. They limit different things.
From the PHB, page 195:
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
So you're not halving your jump distance, you're halving your movement speed. And you have to use that for jumping (PHB page 182):
Either way, each foot you clear on the jump costs a foot of movement.
This means that, for a player whose normal movement speed is 30ft, while grappling he can move 15ft, and that's also the maximum jump height he can achieve. Not even the Jump spell or the Boots of Striding and Springing will let him jump further than that. The effects, however, do let him get around the Standing High Jump penalty.
To summarize, when looking into how high you can possibly jump while grappling, you make the calculations in a few steps:
- Determine your movement speed. This is also your maximum jump height (distance).
- Halve that height if you're grappling a creature that's not two or more sizes smaller than you.
- Determine your base jump distance. Use your total Strength to calculate this (including any bonuses to Strength from items, spells or potions).
- Multiply the base jump distance depending on spells an items you have that increase your jump distance (like the Jump spell).
- If you're making a Standing High Jump, halve the distance from step 4. Otherwise, subtract 10ft from the height in step 2.
- Take the lower number between the numbers that you get in steps 2 and 4 (after modifications in step 5). This number is how high you can jump (and how much movement you'll have to spend to jump the full height).
You can use this little tool I whipped up to test out various combinations.
If you find the maximum jump height (step 2) to be restricting you more than your jump distance achievable (step 4), you can use spells and items that increase your movement speed to propel you further. The Haste spell is good to consider. While it is a substantially higher level spell, it lets you use an extra action to Dash, quadrupling your speed in total. The added AC is a nice touch, too. You can even combine it with the Jump spell, as the latter does not require Concentration.
Using just the effects in your example, however, the highest you can jump is 15ft, using a Standing High Jump. If you somehow throw in a Dash, you can jump twice as much. Note that, if the PC's base Strength is already 18, the Belt does not help increase the jump height in this case.
Keep in mind, though, one litle bit about the High Jump rules:
In some circumstances, your DM might allow you to make a Strength (Athletics) check to jump higher than you normally can.
The rule enables you as a DM to let your player jump beyond his maximum movement speed (this bit of the rule is written directly after the bit about the movement speed restriction). When can that happen, how much of a bonus does the player get, what the DC is and what the possible consequences are (chance of Exhaustion, for example) - all of this is up to you.
A commentary on the Belt of Storm Giant Strength
I personally dislike the use of this item. First of all, it is a legendary item, suitable only for the most glorious of adventurers. Giving this item to a player character that's not very high in level (18 at least, I'd say) would seem overpowered, and frankly uninteresting. Even at that level it might not be as good as it sounds. Surely, it's a fantastic item, but it makes all the precious Ability points a grappler had spent on Strength a waste. On the other hand, having the belt in possession early and spending the points somewhere else (or getting feats), the grappler's main mechanic would now depend entirely on the belt. Losing the belt would thus make the character almost useless. While you might not plan on having it taken away from him in any encounter, the player might become paranoid about it nonetheless.
Alternative to using the belt, using a Haste spell (3rd lvl) and the Boots of Striding and Springing (uncommon item), the player can reach 21ft with a Strength of 18, and 24ft with a strength of 20 (assuming a base movement speed of 30ft, and making a Dash with the Haste extra action) by making a normal High jump (with a 10ft running start). Throw in a Jump spell (1st lvl) and the High jump goes up to 50ft. Using these three effects, the Belt would only give an improvement of 4ft with a Standing high jump.
If the Haste spell is not available, and using Dash is not an option, then a better combo would be to use Boots of Speed (rare item) and the Jump spell. Just those two bring the High jump up to 20ft, which might be enough. In this case, too, increasing Strength above 18 brings no benefits (though I'm sure it does so in the other aspects of grappling).
Seeing as all the Belts of Giant Strength do provide great bonuses to almost all aspects of grappling, I would award the legendary version at lvl 20 as a special Grappler capstone. Of course, not without effort. It's a nice excuse to lead your players into a perilous dungeon. I would also house-rule that the belt require a Strength of 20 to wear, just so all those Ability points don't feel like they're going to waste. If your player left his strength at 18, though, that might not be necessary.
At lower levels, a potion of Giant Strength (whichever tier you fancy) might be a suitable reward for a heroic deed, and since it's only temporary it wouldn't break anything (including fun).