Hmm. Maybe we’re thinking of different things.
I was thinking of a place that cut hair (for men and women). There may be several members of staff, each with their own chair or area. Some may specialise in men’s (or women’s) hair but they will all cut all styles and be part of the same shop. This seems like shop=hairdresser to me.
Perhaps you’re thinking of something different? i.e., almost like two different shops inside one?