The author obtains necessary conditions when \(\max \{\lambda(f), \lambda(f^{-1})\}< \sigma(e^P)\), where \(f(z)\) is a meromorphic solution to higher-order differential equations \[ y^{(k)}+ Q_{k-2}(z) y^{(k-2)}+\cdots+ Q_1(z)y'+ (Q(z)e^{P(z)}+ Q_0(z))y=0. \] The results not only generalize corresponding results of S. Bank, I. Laine, J. K. Langley, Y. M. Chiang, S. Wang and the present author, but also show that the class of coefficients and the oddity and evenness of \(k\) play an important role. In addition, two examples are constructed which fully illustrate the results.