If there was no other context, only the past simple would be possible and the author would have made a mistake. As the sentence finishes after the past perfect clause, I suspect a mistake. This would work, for example: "Having started off in mist-like fashion, the drizzle had moved up a gear by the time he left home to go to work."

Elizabeth Young - 2002 - ‎'A Promising Man' Having started off in mistlike fashion, the drizzle had moved up a gear, as if it were thinking about turning to proper rain, instead.