TE modes have no electric field in the direction of propagation and the TM mode has no magnetic field in the direction of propagation. Generally, the field should be perpendicular to the direction of propagation. Is it possible to have a field distribution in the direction of propagation of an electromagnetic wave?
Yes. One thing that may help you understand this more intuitively is to not think about guided modes like plane waves. Guided modes are more complex. They are more like a sum of at least two waves at some angle "bouncing" back and forth inside the waveguide or transmission line. Further, the field is interacting with whatever structures the waveguide or transmission line has. All of this can cause field components to be in the longitudinal direction.
Hybrid modes are common solutions of Maxwell equations in waveguides with inhomogeneous dielectric. For example, the quasi-TEM mode of microstrip line or the hybrid modes of the optical fibers.
Whenever you need the longitudinal components of both fields to fulfil boundary conditions across longitudinal discontinuities in the dielectric, hybrid modes will likely arise as solutions of Maxwell equations.
An example of transversally inhomogeneous guiding structure that doesn't support hybrid modes is the dielectric slab waveguide. For its special characteristics - it is a 2D waveguide - It only support TE and TM modes.
More on the modal analysis topic:
C.A. Balanis, "Advanced Engineering Electromagnetics", John Wiley, 1989
Regarding Optical Fibers, including modal analysis
J.M. Senior, "Optical Fiber Communications", Prentice Hall, 1985.
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