  Various samples of sodium cobalt oxyhydrate with relatively large amounts of
Na$^{+}$ ions were synthesized by a modified soft-chemical process in which a
NaOH aqueous solution was added in the final step of the procedure. From these
samples, a superconducting phase diagram was determined for a section of a
cobalt valence of $\sim$+3.48, which was compared with a previously obtained
one of $\sim$+3.40. The superconductivity was significantly affected by the
isovalent exchanger of Na$^{+}$ and H$_{3}$O$^{+}$, rather than by variation of
Co valence, suggesting the presence of multiple kinds of Fermi surface.
Furthermore, the high-field magnetic susceptibility measurements for one sample
up to 30 T indicated an upper critical field much higher than the Pauli limit
supporting the validity of the spin-triplet pairing mechanism.
