The direct laser acceleration (DLA) of electrons in a laser wakefield accelerator (LWFA) operating in the forced or quasi-blowout regimes has been investigated through experiment and simulation. When there is a significant overlap between the trapped electrons and the drive laser in a LWFA cavity, the resulting electrons can gain energy from both the LWFA and the DLA mechanisms. Experimental work investigates the properties of the electron beams produced in a LWFA with ionization injection by dispersing those beams in the direction perpendicular to the laser polarization. These electron beams show certain spectral features that are characteristic of DLA. These characteristic features are reproduced using particle-in-cell simulations, where particle tracking was used to elucidate the roles of LWFA and DLA to the energy gain of the electrons in this experimental regime and to demonstrate that such spectral features are definitive signatures of the presence of DLA in LWFA.