We report the magnetic-field amplitude and field-angle dependence of the superconducting onset temperature Tconset of the organic superconductor (TMTSF)2ClO4 in magnetic fields H accurately aligned to the conductive ab' plane. We revealed that the rapid increase of the onset fields at low temperatures occurs both for H || b' and H || a, irrespective of the carrier confinement. Moreover, in the vicinity of the Pauli-limiting field, we report a shift of a principal axis of the in-plane field-angle dependence of Tconset. This feature may be related to an occurrence of Fulde-Ferrell-Larkin-Ovchinnikov phases.