The ultrafast and ultracold electron source, based on laser cooling and trapping of an atomic gas and its subsequent near-threshold photoionization, is capable of generating electron bunches with a high transverse brightness at energies of roughly 10 keV. This paper investigates the possibility of increasing the range of applications of this source by accelerating the bunch using radio-frequency electromagnetic fields. Bunch energies of $\sim35$ keV are measured by analyzing the diffraction patterns generated from a mono-crystalline gold sample. Further analysis points to a largely preserved normalized transverse emittance during acceleration.