The Part 61 FAQ which, correct or not, should be used as guidance by all DPEs seems to say no. (i think it's wrong, but no one put me in charge). The FAQ, through a number of answers, takes the position that (1) the hood work done for the private is "flight solely by reference to instruments", not "instrument training" (that part does make some sense) and (2) that the private tasks were therefore not "actual or simulated instrument time on the areas of operation of this section..." [61.65(d)(2)].
Here's one of the FAQ in the context of a commercial pilot who is going for the instrument rating. Note the last sentence:
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QUESTION: An applicant holds a Commercial certificate (ASEL) and is now working towards an Instrument-Airplane rating which requires at least 40 hours of actual or simulated instrument time as required by 61.65(d)(2). If the applicant has already obtained some actual or simulated instrument time on the areas of operation of covered in § 61.65 but did so in the course of obtaining his Private/Commercial certificate, do those hours count in meeting the requirements of § 61.65(d)(2)?
ANSWER: Ref. § 61.65(d)(2); I can't give you a yes or no straight answer to your question. It depends. The flight instructor is going to have to review the applicant's training records and see whether the training received is equivalent and creditable to the training required by § 61.65(c). But again, it has to have been accomplished after the applicant received his or her Private Pilot Certificate. Because the training required for the Private Pilot applicant by § 61.109(a)(3) is not “instrument training.”
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I have a feeling that not too many DPE actually follow this FAQ since (a) the "rule" is buried in a bunch of them, (b) you have to connect the dots to really see what Lynch is saying, and (c) the reasoning is bogus..