I am trying to find an hyperelliptic curve (say of genus 2) together with an endomorphism $\phi$ of its Jacobian with the property that it sends Mumford reduced divisors of the exceptional form $P-\mathcal{O}$ to divisors of the same form (and not $P+Q-2\mathcal{O}$).

For instance, if $f$ is a morphism from the hyperelliptic curve to itself, then $\phi=f_*$ will have this property. However, such a $f$ must be an automorphism, and therefore $\phi$ has finite order.

My question is: can I find such a $\phi$ which is not a root of unity?