  We theoretically investigate strong-coupling properties of an ultracold Fermi
gas with an orbital Feshbach resonance (OFR). Including tunable pairing
interaction associated with an OFR within the framework of the strong-coupling
theory developed by Nozi\`eres and Schmitt-Rink (NSR), we examine the
occupation of the closed channel. We show that, although the importance of the
closed channel is characteristic of the system with an OFR, the occupation
number of the closed channel is found to actually be very small at the
superfluid phase transition temperature $T_{\rm c}$, in the whole BCS
(Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover region,
when we use the scattering parameters for an ultracold $^{173}$Yb Fermi gas.
The occupation of the closed channel increases with increasing the temperature
above $T_{\rm c}$, which is more remarkable for a stronger pairing interaction.
We also present a prescription to remove effects of an experimentally
inaccessible deep bound state from the NSR formalism, which we meet when we
theoretically deal with a $^{173}$Yb Fermi gas with an OFR.
