We present the median-stacked Lyman-α surface brightness profile of 968 spectroscopically selected Lyman-α emitting galaxies (LAEs) at redshifts 1.9<z<3.5 in the early data of the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). The selected LAEs are high-confidence Lyman-α detections with large signal-to-noise ratios observed with good seeing conditions (point-spread-function full-width-at-half-maximum <1.4 arcsec), excluding active galactic nuclei (AGN). The Lyman-α luminosities of the LAEs are $10^{42.4}$−$10^{43}$ erg/s. We detect faint emission in the median-stacked radial profiles at the level of $(3.6\pm 1.3) \cdot 10^{-20}$ erg/s/cm$^{-2}$/arcsec$^{-2}$ from the surrounding Lyman-α halos out to $r\approx 160$ kpc (physical). The shape of the median-stacked radial profile is consistent at $r<80$ kpc with that of much fainter LAEs at 3<z<4 observed with the Multi Unit Spectroscopic Explorer (MUSE), indicating that the median-stacked Lyman-α profiles have similar shapes at redshifts 2<z<4 and across a factor of 10 in Lyman-α luminosity. While we agree with the results from the MUSE sample at $r<80$ kpc, we extend the profile over a factor of two in radius. At $r>80$ kpc, our profile is flatter than the MUSE model. The measured profile agrees at most radii with that of galaxies in the Byrohl et al. (2021) cosmological radiative transfer simulation at z=3. This suggests that the surface brightness of a Lyman-α halo at $r\leq 100$ kpc is dominated by resonant scattering of Lyman-α photons from star-forming regions in the central galaxy, whereas at $r>100$ kpc it is dominated by photons from galaxies in surrounding dark matter halos.