In non-Kramers Kondo systems with quadrupolar degrees of freedom, an ordinary magnetic Kondo effect can compete with the quadrupolar Kondo effect. We discuss such competition keeping PrT2Zn20 (T = Ir, Rh) and PrT2Al20 (T = V, Ti) in mind, where the Γ3 non-Kramers crystalline-electric-field (CEF) doublet ground state is realized in a Pr3+ ion with a (4f)2 configuration under cubic symmetry. The quadrupolar Kondo effect can be described by the two-channel Kondo model, which leads to the local non-Fermi-liquid (NFL) ground state, while the magnetic Kondo effect favors the ordinary local Fermi-liquid (FL) ground state. On the basis of the minimal extended two-channel Kondo model including the magnetic Kondo coupling as well, we investigate the competition and resulting thermodynamics, and orbital/magnetic and single-particle excitation spectra by Wilson's numerical renormalization group (NRG) method. There is a first-order transition between the NFL and FL ground states. In addition to these two states, the alternative FL state accompanied by a free magnetic spin appears in the intermediate temperature range, which eventually reaches the true NFL ground state, as a consequence of the stronger competition between the magnetic and quadrupolar Kondo effects. In this peculiar state, the magnetic susceptibility shows a Curie-like behavior, while the orbital fluctuation exhibits the FL behavior. Moreover, the single-particle spectra yield a more singular behavior. Implications to the Pr 1-2-20 systems are briefly discussed.