We present an electron injection scheme for plasma wakefield acceleration. The method is based on a recently proposed technique of fast electron generation via laser-solid interaction: a femtosecond laser pulse with the energy of tens of mJ hitting a dense plasma target at 45^{∘} angle expels a well collimated bunch of electrons and accelerates these close to the specular direction up to several MeVs. We study trapping of these fast electrons by a quasilinear wakefield excited by an external beam driver in a surrounding low density plasma. This configuration can be relevant to the AWAKE experiment at CERN. We vary different injection parameters: the phase and angle of injection, the laser pulse energy. An approximate trapping condition is derived for a linear axisymmetric wake. It is used to optimize the trapped charge and is verified by three-dimensional particle-in-cell simulations. It is shown that a quasilinear plasma wave with the accelerating field ∼ 2.5 GV/m can trap electron bunches with ∼ 100 pC charge, ∼60μm transverse normalized emittance and accelerate them to energies of several GeV with the spread ≲ 1% after 10 m..