The Tabaxi racial trait "Feline Agility" (Volo's Guide to Monsters, p. 115) says:
When you move on your turn in combat, you can double your speed until the end of the turn.
The Ring of Jumping (from the Basic Rules) says:
While wearing this ring, you can cast the jump spell from it as a bonus action at will, but can target only yourself when you do so.
The jump spell (Basic Rules) says:
[your] jump distance is tripled
How do those combine?
As a specific example, Tabitha the Tabaxi Bard, strength of 10, is 30 feet from, for instance, a hell hound (bad dog!). How could she use jump from the ring to get the jump on that damned dog?
Best Answer
Your speed and the distance you can jump are related, but separate.
Base speed
The basic rules define a creature's speed like this:
Other actions and features key off your speed to determine different modes of movement, or the way your movement works in combat. Most importantly, the description of movement in combat states:
Base jumping distance
The rules on jumping are provided here:
As you can see, jumping is not separate from movement; it uses your movement.
The jump spell
The jump spell (and the ring of jumping that you can cast it from) doesn't affect your speed at all. It solely triples your jump distance. What that means is:
Note that the spell does not affect your speed at all. Jeremy Crawford clarifies here and here on Twitter that your jump distance is still limited by how far you can move (though you can take the Dash action to increase your available movement for the turn).
Feline Agility
This one's easy. It simply doubles your speed until the end of the turn. Nothing more, nothing less. Anything that keys off your speed, including the distance you can move in a turn, is now keying off your doubled speed until the end of the turn. (Note that the trait does specify that once you use it to double your speed, you can't use it again until you move 0 feet on one of your turns.)
Tabitha the Tabaxi Bard, with jump and Feline Agility active
You specified this example:
If she has a Strength score of 10, her running long jump distance would normally be 10 feet (i.e. with a running start of 10 feet before she jumps); with a standing long jump, she could only jump half that distance, for a total of 5 feet. No matter how far she jumps, it uses her movement to do so (in addition to any movement needed for the running start if applicable).
With the jump spell, her jump distance is tripled. She could now make a running long jump of 30 feet; however, she would need a running start of 10 feet before she jumps to get that far, using a total of 40 feet of movement. (Without a running start, she could only jump 15 feet.)
However, tabaxi have a regular walking speed of 30 feet, so normally she'd need to take the Dash action (which adds an amount equal to your speed to your movement) in order to make a maximum-length running long jump. Thankfully, Tabaxi also have the Feline Agility trait, which allows them to choose to double their speed for one turn (and they can only use the ability to do so again after they spend a turn not using their movement).
By activating the Feline Agility trait, Tabitha has a speed of 60 feet for a turn; using her Ring of Jumping to cast jump on herself as a bonus action, her jump distance is tripled. She would thus be able to get a running start of 10 feet, jump a maximum of 30 feet (i.e. 3 * Str score), and still have 20 feet of movement and an action left on her turn to use as she wishes.
(That said, jumping across regular terrain doesn't really accomplish much, except maybe looking cool. She'd only really need to make that sort of horizontal jump to get over a crevasse/pit or environmental hazard.)