We report an observation of smooth spin-reorientation transitions from in-plane to normal-to-plane direction with increasing Pd overlayer thickness in CoxPd1-x alloy films, showing characteristics of a second-order transition (SOT). The magnetoelastic coupling term Gamma plays a crucial role in such behavior via its contribution to the fourth- as well as second-order anisotropy constants. The nature of the SOT is characterized by -Gamma(4)/Gamma(2)=0.12-0.47 determined for x=0.25-0.40 in conjunction with lattice-misfit strains. The controlling spin orientation or anisotropy axis via magnetoelastic coupling without applying a magnetic field may be applicable to miniatured memory and sensor devices.