SN 2016gkg is a nearby SN IIb discovered shortly after explosion. Like several other Type IIb events with early-time data, SN 2016gkg displays a double-peaked light curve, with the first peak associated with the cooling of a low-mass extended progenitor envelope. We present unprecedented intranight-cadence multi-band photometric coverage of the first light curve peak of SN 2016gkg obtained from the Las Cumbres Observatory Global Telescope network, the Asteroid Terrestrial-impact Last Alert System, the Swift satellite, and various amateur-operated telescopes. Fitting these data to analytical shock-cooling models gives a progenitor radius of ∼40-150 with ∼2-40 × 10−2 of material in the extended envelope (depending on the model and the assumed host-galaxy extinction). Our radius estimates are broadly consistent with values derived independently (in other works) from HST imaging of the progenitor star. However, the shock-cooling model radii are on the lower end of the values indicated by pre-explosion imaging. Hydrodynamical simulations could refine the progenitor parameters deduced from the shock-cooling emission and test the analytical models.