Since the launch of the Fermi Gamma-ray Space Telescope on 2008 June 11, the Large Area Telescope (LAT) instrument has firmly detected more than 20 gamma-ray bursts (GRBs) with high-energy photon emission above 100 MeV. Using the matched filter technique, three more GRBs have also shown evidence of correlation with high-energy photon emission as demonstrated by Akerlof et al. In this paper, we present another GRB, GRB 081006A, unambiguously detected by the matched filter technique. This event is associated with more than 13 high-energy photons above 100 MeV. The likelihood analysis code provided by the Fermi Science Support Center generated an independent verification of this detection using a comparison of the test statistics value with similar calculations for random LAT data fields. We have performed detailed temporal and spectral analysis of photons from 8 keV up to 0.8 GeV from the Gamma-ray Burst Monitor and the LAT. The properties of GRB 081006A can be compared to those of the other two long-duration GRBs detected at similar significance, GRB 080825C and GRB 090217A. We find that GRB 081006A is more similar to GRB 080825C with comparable appearances of late high-energy photon emission. As demonstrated previously, there appears to be a surprising dearth of faint LAT GRBs, with only one additional GRB identified in a sample of 74. In this unique period when both Swift and Fermi are operational, there is some urgency to explore this aspect of GRBs as fully as possible.