This answer below, in the corollary, is roughly the same as in my comment. The missing ingredient is the fact that every fppf cover of a quasi-compact scheme is "refined" by an fppf cover that is quasi-finite.  This is a Bertini hyperplane theorem.

Let $X$ be a Noetherian scheme.  Let $p:Y\to X$ be an fppf morphism.

<B>Proposition.</B>  There exists a morphism $i:Z\to Y$ that is a disjoint union of locally closed immersions such that the composition $p\circ i$ is fppf and quasi-finite.

<B>Proof.</B>  This argument is essentially a Bertini hyperplane argument. First of all, up to replacing $Y$ by a disjoint union of open affine subschemes, assume that $Y$ is affine.  Denote by $Y_{\text{equi}}\subset Y$ the maximal open subset on which $p$ is (locally) equidimensional, i.e., at every point of every fiber of $p$, the irreducible components of the fiber containing the point all have equal dimensions.  The restriction of $p$ to $Y_{\text{equi}}$ is still surjective.  Thus, without loss of generality, assume that $p$ is (locally) equidimensional.  Then on every connected component $Y_\alpha$ of $Y$, the fiber dimension of $p$ is constant.  We construct $Z$ as a disjoint union of schemes $Z_\alpha$ where $i_\alpha:Z_\alpha\to Y_\alpha$ is a disjoint union of locally closed immersions such that $p_\alpha\circ i_\alpha$ is a quasi-finite, flat morphism with image equal to the image of $p_\alpha$.

Thus, without loss of generality, assume that $Y$ is equidimensional of dimension $d$.  The proof of the existence of $i:Z\to Y$ as above is by induction on $d$.  If $d$ equals $0$, then define $Z$ to equal $Y$.  Thus, by way of induction, assume that $d>0$ and assume the result is proved for $d-1$.  

Let $e:Y\hookrightarrow \mathbb{A}^n_X$ be a closed immersion of $X$-schemes.  For every point in the image of $p$, $x:\text{Spec}\ k \to X,$ the fiber $Y_x=p^{-1}(x)$ has only finitely many associated points.  For every linear polynomial $t_x$ in the coordinate ring $k[\mathbb{A}^n_k]=k[t_1,\dots,t_n]$ that vanishes at none of these finitely many associated points, multiplication by $t_x$ is injective on $\mathcal{O}_{Y_p}$.  Let $t\in\mathcal{O}_{X,x}[t_1,\dots,t_n]$ be an element mapping to $t_x$. Denote by $H$ the zero scheme of $t$.  

By the local flatness criterion, the intersection $H\cap Y$ is flat at $x$, cf. Theorem 22.5, p. 176, "Commutative ring theory", H. Matsumura, Cambridge studies in mathematics, vol 8.  By the usual arguments, there exists an open affine neighborhood of $x$ such that $t$ lifts to a section of the structure sheaf of $\mathbb{A}^n_X$ over this open.  Define $H$ to be the zero scheme of $t$ on this open.  By openness of the flat locus, etc., there exists an open subscheme $U$ of $Y$ containing $H\cap Y_x$ such that $H\cap U$ is flat over $X$.  Choosing $t_x$ so that $H\cap Y_x$ is nonempty, $U\cap H \to X$ has image containing $p$, and it is equidimensional of dimension $d-1$ (this last by Krull's Hauptidealsatz).  

Thus, by the induction hypothesis applied to $U\cap H \to X$, there exists a locally closed immersion $i_x:Z_x\to U\cap H$ such that $p\circ i_x$ is quasi-finite and flat with image $V_x$ containing $x$.  As we vary $x$, the open images $V_x$ cover $X$.  Since we assumed that $X$ is quasi-compact, finitely many of these opens suffice to cover $X$.  Define $i:Z\to Y$ to be the disjoint union of these finitely many locally closed immersions $i_x$.  Since the composition $p\circ i$ is quasi-finite and flat on each of the finitely many connected components, it is quasi-finite and flat.  By construction, the image equals the image of $p$.  Thus, the lemma is proved by induction on $d$. <B>QED</B>

<B>Lemma.</B>  Let $q:Z\to X$ be a quasi-finite, flat morphism that is surjective.  There exists a dense open subscheme $j:V\hookrightarrow X$ and a positive integer $n$ such that the kernel of the pullback homomorphism $q^*:\text{Pic}(X)\to \text{Pic}(Z)$ is contained in the kernel of the homomorphism $\text{Pic}(X)\xrightarrow{j^*}\text{Pic}(V)\xrightarrow{n\cdot -} \text{Pic}(V).$

<B>Proof.</B>
For every finitely presented, quasi-finite morphism $q:Z\to X$, there exists a dense open subset $V\subset X$ such that $q^{-1}(V)\to V$ is finite.  There exists a norm map, $$\text{Nm}_q : \text{Pic}(q^{-1}(V))\to \text{Pic}(V), \ \ \mathcal{L} \mapsto \text{det}(q_*\mathcal{L})\otimes \text{det}(q_*\mathcal{O})^{\vee}.$$  This is described in detail in Mumford's <I>Lectures on curves on an algebraic surface.</I>  For every connected component $V_\alpha$ of $V$, the morphism $q$ has constant finite degree $m$ over that component. For every invertible sheaf $\mathcal{M}_\alpha$ on $V_\alpha$, $\text{Nm}_q(q^*\mathcal{M}_\alpha)$ equals $\mathcal{M}_\alpha^{\otimes m}$.  Thus, defining $n$ to equal the least common multiple of these integers $m$, for every invertible sheaf $\mathcal{M}$ on $V$, if $q^*\mathcal{M}$ is trivial, then also $\mathcal{M}^{\otimes n}$ is trivial.  Thus, for an invertible sheaf $\mathcal{M}$ on $X$, if $q^*\mathcal{M}$ is trivial, then also $j^*\mathcal{M}^{\otimes n}$ is trivial. <B>QED</B>

<B>Corollary.</B>  Let $k$ be an algebraically closed field.  For every smooth, projective $k$-scheme $X$ such that $\text{Pic}^0_{X/k}$ has positive dimension, for every fppf morphism $p:Y\to X$, the image of $\text{Pic}^0_{X/k}(k)$ in $\text{Pic}(Y)$ is an infinitely generated, divisible group.

<B>Proof.</B> The group of $k$-points $\text{Pic}^0_{X/k}(k)$ is divisible, has finite $n$-torsion for every integer $n$, and is infinitely generated.  The proposition precisely says that the image of $p^*$ is still infinitely generated.  

By the proposition, there exists a morphism $i:Z\to Y$ such that $q=p\circ i:Z\to X$ is fppf and quasi-finite.  Since the image of $p^*$ is intermediate between $\text{Pic}^0_{X/k}(k)$ and its image under $q^*$, it suffices to prove that the image under $q^*$ is infinitely generated.  By the lemma, there exists a dense open subset $j:V\hookrightarrow X$ and an integer $n$ such that the kernel of $q^*$ is contained in the kernel of $$\text{Pic}^0_{X/k}(k)\xrightarrow{j^*}\text{Pic}(V)\xrightarrow{n\cdot -}\text{Pic}(V).$$ 

The kernel of the restriction map $j^*:\text{Pic}(X)\to \text{Pic}(V)$ is generated by the finitely many irreducible components of $X\setminus V$ that have codimension $1$ in $X$.  Thus, the intersection of this kernel with the subgroup $\text{Pic}^0_{X/k}(k)$ is a subgroup of a finitely generated group, hence it is also finitely generated.  The quotient of $\text{Pic}^0_{X/k}(k)$ by this finitely generated subgroup is still divisible and it is still infinitely generated.  Moreover, it still has finite $n$-torsion for every integer $n$: the inverse image in $\text{Pic}^0_{X/k}(k)$ of this $n$-torsion is an extension of a subgroup of the finitely generated kernel by the $n$-torsion of $\text{Pic}^0_{X/k}(k)$.  Thus the image is an $n$-torsion group that is finitely generated (the image of a finitely generated group), hence it is finite.  So, for each integer $n$, the quotient of this group by its $n$-torsion is still an infinitely generated, divisible group. <B>QED</B>