Laser-heater systems have been demonstrated to be an important component for the accelerators that drive high gain free electron laser (FEL) facilities. These heater systems suppress longitudinal microbunching instabilities by inducing a small and controllable slice energy spread to the electron beam. For transversely uniform heating, the energy spread augmentation is characterized by a non-Gaussian distribution. In this Letter, we demonstrate experimentally that in addition to suppression of the microbunching instability, the laser heater-induced energy distribution can be preserved to the FEL undulator entrance, significantly impacting the performance of high-gain harmonic generation (HGHG) FELs, especially at soft x-ray wavelengths. In particular, we show that the FEL intensity has several local maxima as a function of the induced heating caused by the non-Gaussian energy distribution together with a strong enhancement of the power at high harmonics relative to that expected for an electron beam with an equivalent Gaussian energy spread at an undulator entrance. These results suggest that a single stage HGHG FEL can produce scientifically interesting power levels at harmonic numbers m ≥ 25 and with current seed laser technology could reach output photon energies above 100 eV or greater.