We present results of a search for evidence of reentrance at high magnetic field of the superconducting state in the quasi-one dimensional organic superconductor (TMTSF) 2ClO 4, for a specific orientation of the field at low temperature. This search was motivated by the theoretical work of Lebed' and of Dupuis & Montambaux, and in part by that of Teŝanovic and coworkers, who predict novel magnetic field-temperature phase diagrams in quasi-1-D and quasi-2-D superconductors in a strong magnetic field. In the quasi-1-D case, a field directed both normal to the chains and along the layers is proposed to induce exotic oscillations of the critical temperature at high field, while for the 2-D case, quantum effects associated with Landau quantization lead to similar predictions. We have investigated the superconducting transition in a quasi-1-D candidate material, (TMTSF) 2ClO 4, to 0.06 K with field oriented along the b′-axis (±0.02°). From mostly constant field temperature sweeps, we find positive curvature in H c2(T) below about T c/2. In addition, a resistivity decrease is observed to persist to 7 T, the highest field employed, with a slightly positive slope ∂H/∂T above 5T. The possibility of these features being representative of high field superconductivity is discussed.