For a hyperbolic second-order differential operator $P$, we study the relations between the maximal Gevrey index for the strong Gevrey well-posedness and some algebraic and geometric properties of the principal symbol $p$. If the Hamilton map $F_p$ of $p$ (the linearization of the Hamilton field $H_p$ along double characteristics) has nonzero real eigenvalues at every double characteristic (the so-called effectively hyperbolic case), then it is well known that the Cauchy problem for $P$ is well posed in any Gevrey class $1\le s<+\infty$ for any lower-order term. In this paper we prove that if $p$ is noneffectively hyperbolic and, moreover, such that $Ker{F}_p^2\cap Im{F}_p^2\ne \left\{0\right\}$ on a $C^{\infty }$ double characteristic manifold $\Sigma$ of codimension $3$, assuming that there is no null bicharacteristic landing $\Sigma$ tangentially, then the Cauchy problem for $P$ is well posed in the Gevrey class $1\le s<4$ for any lower-order term (strong Gevrey well-posedness with threshold $4$), extending in particular via energy estimates a previous result of Hörmander in a model case.