  Gamma-ray bursts produce afterglows that can be observed across the
electromagnetic spectrum and can provide insight into the nature of their
progenitors. While most telescopes that observe afterglows are designed to
rapidly react to trigger information, the Transiting Exoplanet Survey Satellite
(TESS) continuously monitors sections of the sky at cadences between 30 minutes
and 200 seconds. This provides TESS with the capability of serendipitously
observing the optical afterglow of GRBs. We conduct the first extensive search
for afterglows of known GRBs in archival TESS data reduced with the TESSreduce
package, and detect 11 candidate signals that are temporally coincident with
reported burst times. We classify 3 of these as high-likelihood GRB afterglows
previously unknown to have been detected by TESS, one of which has no other
afterglow detection reported on the Gamma-ray Coordinates Network. We classify
5 candidates as tentative and the remainder as unlikely. Using the afterglowpy
package, we model each of the candidate light curves with a Gaussian and a top
hat model to estimate burst parameters; we find that a mean time delay of
$740\pm690\,$s between the explosion and afterglow onset is required to perform
these fits. The high cadence and large field of view make TESS a powerful
instrument for localising GRBs, with the potential to observe afterglows in
cases when no other backup photometry is possible.
