As written, any target* of the Jump spell may immediately jump 30ft. Targets of the 3rd level Jump spell can then repeat the effect using the Leap Action.
From Heightened Spells:
In addition, many spells have additional specific benefits when they are heightened, such as increased damage. These extra benefits are described at the end of the spell’s stat block. Some heightened entries specify one or more levels at which the spell must be prepared or cast to gain these extra advantages. Each of these heightened entries states specifically which aspects of the spell change at the given level.
The Jump spell reads
Your legs surge with strength, ready to leap high and far. You jump 30 feet in any direction without touching the ground. You must land on a space of solid ground within 30 feet of you, or else you fall after using your next action.
Heightened (3rd) The range becomes touch, the target changes to one touched creature, and the duration becomes 1 minute, allowing the target to jump as described whenever it takes the Leap action.
The Heightened entry doesn't include "instead" or any such language, so the modifications to targeting and duration do not alter the base benefit of the spell. Deciding that there is no initial benefit is probably a reasonable table decision, but in such a case I would recommend that it applies equally to the caster themselves and other targets they may pick.
*Technically, an overly strict RAW is that the spellcaster gains the one-time benefit of Jump while the target gains the continuous effect. I don't think this makes any sense and should fall under making a ruling that seems correct over the literal meaning of the words.