Suppose that $G = \langle X \rangle$ is a $\delta$-hyperbolic group, i.e. all geodesic triangles are $\delta$-thin (the inverse image of a point under the projections onto a tripod has diameter bounded by $\delta$).
It is a standard result that every finite subgroup of $G$ is conjugate to a subgroup contained in the ball of radius $2 \delta +1$ around the identity. It follows that every finite subgroup of $G$ is of size at most $|X|^{2\delta + 1} + 1$.
Can this result be improved if we only considered cyclic subgroups, i.e. can we get a better bound on the maximal order of a torsion element than $|X|^{2\delta + 1} + 1$?
EDIT: Following the discussion in the comments, does the situation change if we assumed the group $G$ to be one-ended?