The relativistic dynamics of an electron accelerated in a cylindrical cavity mode TE113 in the presence of a static inhomogeneous magnetic field is studied. This type of acceleration is known as Spatial AutoResonance Acceleration (SARA). The magnetic field profile is such that it keeps the phase difference between the electric microwave field and the electron velocity vector within the acceleration phase band. We study the dynamic of the electron through simulations of the relativistic Newton-Lorentz equation. Numerical experiments with TE113 cylindrical microwave fields of a frequency of 2.45GHz and an amplitude of the order of 7kV/cm show that it is possible accelerate the electrons up to energies of the order of 300keV. This energy is about of 30% higher than those obtained in previous studies by using the TE112 mode.