The only time I did that, the company I was leaving begged me to take one of their corp laptops with me so I could offer "consulting" to them on the side. I never called them on their plausible deniability, I just said I felt that would be a conflict of interest and they shut up real fast.
Can you elaborate on the plausible deniability part.
Was your past employer trying to make best of the situation by hoping to fish for information about your next employer's projects via the consulting gig?
I said "never called them on their plausible deniability" to mean I never pulled off the veil of the conversation, so there's nothing to talk about in front of a judge.
FWIW both the company I was leaving and the company I was joining were startups selling dynamic seat pricing systems to airlines. Your call if that's a conflict of interest :).
Ehmm... offering "consulting" should come with a contract specifying scope of work, data/device access, and expectations.