If a process has several purification steps and the first step fails to meet specifications, but this can be corrected by adjustment in a subsequent step, is an OOS investigation needed for this first step?

An OOS result is an out of specification result for the specification that you set at that first step.
Theoretically you set that at the level at which you think it needs to be investigated. If it exceeds
your specification, regardless of where it is, you should investigate it.