Yes, they stack. The ring mentions it is 'as though the jump spell were cast on them', and the Jump spell in 3.0 gives an unnamed bonus (unusually enough, it doesn't say it's an enhancement bonus; that, I believe, was changed in 3.5?). So technically, you could even craft a custom magic item that gives an enhancement bonus to Jump checks and it would also stack with the Boots of Striding and Springing and the Ring of Jumping.
For even more crazy-ninja-super-leaping, dip 1-4 levels into Psychic Warrior (3.0), and take two copies of the Mental Leap feat. You'll need something like 2 or 3 copies of the Inner Strength feat to meet the power point requirement for a second Mental Leap, but 1 PsyWar level and 1 Inner Strength would give you enough power points to make use of the first Mental Leap feat (3.0 PsyWars get a bonus feat at 1st and another at 2nd, like fighters, and can choose psionic feats instead of fighter feats when desired). Or dip 2 levels into Psychic Warrior and 1 level into Psion, then you shouldn't need any Inner Strength feats if you have really high Dexterity and choose to be a Nomad Psion (and will be able to get 2 Mental Leap feats as PsyWar bonus feats). Otherwise you'll still need 2 Inner Strength feats for the power point requirement of the 2nd Mental Leap feat....
But dang will you be jumpin' around like a crazy, mutant, cyborg jackrabbit on crack.....