Yes, there are.

In fact, in this paper by LeBrun and Mason, they refer to a result by Guillemin 1976 according to which for any odd smooth function $f:S^2\to \mathbb{R}$, there is a one-parameter family $g_t=\exp(f_t)g_0$ of smooth Zoll metrics with $g_0$ the round one and $(df_t/dt)_{t=0}=f$.

If $f$ is not rotationally symmetric, you have examples.

Lebrun and Mason classify instead Zoll projective structures by very nice twistor-like constructions.

ADDED: Guillemin's result is detailed in A. Besse "Manifolds all of whose geodesics are closed", p.126 theorem 4.70.

Yes, there are.

In fact, in this paper by LeBrun and Mason, they refer to a result by Guillemin 1976 according to which for any odd smooth function $f:S^2\to \mathbb{R}$, there is a one-parameter family $g_t=\exp(f_t)g_0$ of smooth Zoll metrics with $g_0$ the round one and $(df_t/dt)_{t=0}=f$.

If $f$ is not rotationally symmetric, you have examples.

Lebrun and Mason classify instead Zoll projective structures by very nice twistor-like constructions.

ADDED: Guillemin's result is detailed in A. Besse "Manifolds all of whose geodesics are closed", p.126 theorem 4.70.