Hyperelliptic $d$-osculating covers and rational surfaces

Let $\mathbb{P}^1$ and $(X,q)$ denote, respectively, the projective line and a fixed elliptic curve marked at its origin, both defined over an algebraically closed field $\mathbb{K}$ of arbitrary characteristic $\emph{\textbf{p}} \neq2$. We will consider all finite separable marked morphisms $\pi :(\Gamma ,p)\rightarrow (X,q)$, such that $\Gamma$ is a degree-$2$ cover of $\mathbb{P}^1$, ramified at the smooth point $p \in \Gamma$. Canonically associated to $\pi$ there is the Abel (rational) embedding of $\Gamma$ into its \emph{generalized Jacobian}, $A_p: \Gamma \to Jac\,\Gamma$, and $\{0\} \subsetneq V^1_{\Gamma,p}...\subsetneq V ^g_{\Gamma,p}$, the flag of hyperosculating planes to $A_p(\Gamma)$ at $A_p(p)\in Jac\,\Gamma$ (cf. \textbf{2.1. & 2.2.}). On the other hand, we also have the homomorphism $\iota_\pi: X \to \Jac\,\Gamma$, obtained by dualizing $\pi$. There is a smallest positive integer $d$ such that the tangent line to $\iota_\pi( X)$ is contained in $V^d_{\Gamma,p}$. We call it \emph{the osculating order} of $\pi$. Studying, characterizing and constructing those with given \textit{osculating order} $d$ but maximal possible arithmetic genus, is one of the main issues. The other one, to which the first issue reduces, is the construction of all rational curves in a particular anticanonical rational surface associated to $X$ (i.e.: a rational surface with an effective anticanonical divisor).