In this paper, the authors consider the Cauchy problem \[ \begin{cases} P(t,\partial_t,\partial_x)u(t,x)=0,\quad(t,x)\in[0,T]\times\mathbb R\\ \partial_t^ju(0,x)=u_j(x),\quad x\in\mathbb R,\quad j=0,...,m-1 \end{cases}\tag{CP} \] where \(P\) is a differential operator of order \(m\) with respect to \(t\) written in the form \[P(t,\partial_t,\partial_x)=L(t,\partial_t,\partial_x)+M(t,\partial_t,\partial_x)\] with \begin{itemize} \item \(L(t,\partial_t,\partial_x)=\partial_t^m+\displaystyle\sum_{j=1}^{m}a_j(t)\partial_t^{m-j}\partial_x^j\), \item \(M(t,\partial_t,\partial_x)=\displaystyle\sum_{j+j\leq m-1}b_{j,h}(t)\partial_t^{j}\partial_x^h,\) \end{itemize} the coefficients \(a_j(t)\) belonging to \(\mathcal C^{\infty}([-T,T])\), and the coefficients \(b_{j,h}(t)\) belonging to \(L^{\infty}([-T,T])\). It is well-known that, in order the problem (CP) to be well-posed in \(\mathcal C^{\infty}\) and in Gevrey spaces, the operator \(P\) needs to be hyperbolic, that is the solutions in \(\tau\) of the characteristic equation \(L(t,\tau;1)=0\) (= the characteristic roots) are all real [\textit{P. D. Lax}, Duke Math. J. 24, 627--646 (1957; Zbl 0083.31801); \textit{S. Mizohata}, J. Math. Kyoto Univ. 1, 109--127 (1961; Zbl 0104.31903); \textit{T. Nishitani}, J. ibid. 18, 509--521 (1978; Zbl 0402.35093)]. In the special case where \(P\) is strictly hyperbolic, that is the characteristic roots are real and distinct, then (CP) is well-posed in \(\mathcal C^{\infty}\) and in all Gevrey spaces. On the other hand, if \(P\) is weakly hyperbolic, that is the characteristic roots are real and may coincide, then Bronšteǐn proved in [\textit{M. D. Bronshtejn}, Tr. Mosk. Mat. O.-va 41, 83--99 (1980; Zbl 0468.35062)] that (CP) is well-posed in the \(s\)-Gevrey space for any \(s\in]1,r/(r-1)[\), where \(r\) stands for the largest multiplicity of the characteristic roots, the bound \(r/(r-1)\) being sharp in general. In this paper, the authors are interested in finding conditions on the principal symbol \(L\) so that (CP) is well-posed in some \(s\)-Gevrey space with \(s>r/(r-1)\), for any lower order operator \(M\), generalizing thus the result recently obtained by \textit{T. Nishitani} [Osaka J. Math. 54, No. 2, 383--408 (2017; Zbl 1383.35118)].