How much steric hindrance does a phenyl group offer?
despite being what I thought was a rather large substituent


I agree, generally speaking the phenyl group is considered a rather large substituent.

One metric commonly used to compare "steric size" is the cyclohexane A-Value.  In the cyclohexane ring system, bulky substituents prefer the equatorial position rather than the axial position. This is due to steric (this can also be described in electronic terms since it is due to the repulsion between interpenetrating electron clouds) destabilization. For example, in methylcyclohexane, the methyl group can exist in an axial or equatorial position. In the axial position, it is destabilized by the repulsive steric (electronic) interactions it has with the axial hydrogens on carbons 3 and 5.



The larger the A-Value, the larger the substituent and the greater its preference for the equatorial location ($\mathrm{\Delta G = -RTlnK_{eq}}$ where $\mathrm{K_{eq}}$ represents the the axial/equatorial ratio at equilibrium).  

Here is a link to a Wikipedia article on A-Values.  Look at the table of A-Values in this article.  A methyl group has an A-Value of 1.7 kcal/mol. a phenyl group has an A-value of 3 kcal/mol.  The only substituent in the table with an A-Value larger than phenyl is the t-butyl substituent with an A-Value > 4 kcal/mol. The phenyl group is a relatively large substituent.