The closer the eccentricity is to 1 without being equal or less, the more "curved" the trajectory is and the nearer its closes approach to the sun will be.
Is this true? I thought it depends on the object’s speed. You can have an object with e=3 have a closer approach to the sun than an object with e=2 if the first object is traveling sufficiently faster.
In other words, at a given perihelion, you can change an object’s e by accelerating or decelerating. Not a orbital mechanics major, just played too much KSP.
Talking about the "speed" is potentially unclear as an object's speed can vary greatly within its orbit.