Of course you can.
What happens if you have a Belt of Battle, a move speed of 40, and you need to go 80 feet this round and still attack something?
Spend your swift to gain a move from the Belt, then use your normal move to get where you want to go. then attack.
Same thing here.
The issue is if beating your movement speed with a jump check is really trivial. Doing a 40 foot high jump would require a Check of 200 (if you got a 20 foot running start or had the right feats/abilities, otherwise it's 400). At level 20 we have 23 ranks, a modifier of at least +10, +4 more from our movement speed, +30 from the Halberd, +30 from a competence item, +10 bonus feet from leaping dragon stance...
That's a bonus of about +99 so far... On a natural 20 we get about 29 feet.