Two radiocarbon (14C) excursions are caused by an increase of incoming cosmic rays on a short time scale found in the Late Holocene (AD 774–775 and AD 993–994), which are widely explained as due to extreme solar proton events (SPE). In addition, a larger event has also been reported at 5480 BC (Miyake et al. 2017a), which is attributed to a special mode of a grand solar minimum, as well as another at 660 BC (Park et al. 2017). Clearly, other events must exist, but could have different causes. In order to detect more such possible events, we have identified periods when the 14C increase rate is rapid and large in the international radiocarbon calibration (IntCal) data (Reimer et al. 2013). In this paper, we follow on from previous studies and identify a possible excursion starting at 814–813 BC, which may be connected to the beginning of a grand solar minimum associated with the beginning of the Hallstatt period, which is characterized by relatively constant 14C ages in the period from 800–400 BC. We compare results of annual 14C measurements from tree rings of sequoia (California) and cedar (Japan), and compare these results to other identified excursions, as well as geomagnetic data. We note that the structure of the increase from 813 BC is similar to the increase at 5480 BC, suggesting a related origin. We also assess whether there are different kinds of events that may be observed and may be consistent with different types of solar phenomena, or other explanations.