Let $p$ be a (large) prime.

Does there exist a function $f\colon\mathbb F_p^\times\to\mathbb F_p$ such that the three sets $$ \{f(z)-z\colon z\in\mathbb F_p^\times\},\ \{f(z)\colon z\in\mathbb F_p^\times\},\ \text{and}\ \{f(z)+z\colon z\in\mathbb F_p^\times \} $$ form a double-cover of $\mathbb F_p$ (in the sense that any element of $\mathbb F_p$ lies in at least two of them)?

More generally,

Do there exist a function $f\colon\mathbb F_p^\times\to\mathbb F_p$ and elements $c_1,c_2,c_3\in\mathbb F_p$ such that the images of the functions $z\mapsto f(z)-c_iz\ (z\in\mathbb F_p^\times)$ double-cover $\mathbb F_p$?

I hardly believe this is a tough problem; rather, I suspect I am overlooking a simple argument.