We performed a search for the K_{L}→π^{0}νν[over ¯] decay using the data taken in 2021 at the J-PARC KOTO experiment. With newly installed counters and new analysis method, the expected background was suppressed to 0.252±0.055_{stat} _{-0.067^{syst}}^{+0.052}. With a single event sensitivity of (9.33±0.06_{stat}±0.84_{syst})×10^{-10}, no events were observed in the signal region. An upper limit on the branching fraction for the decay was set to be 2.2×10^{-9} at the 90% confidence level (C.L.), which improved the previous upper limit from KOTO by a factor of 1.4. With the same data, a search for K_{L}→π^{0}X^{0} was also performed, where X^{0} is an invisible boson with a mass ranging from 1 to 260 MeV/c^{2}. For X^{0} with a mass of 135 MeV/c^{2}, an upper limit on the branching fraction of K_{L}→π^{0}X^{0} was set to be 1.6×10^{-9} at the 90% C.L.