You are definitely entitled to claim the trip home as business mileage. But, read the details below.
You are not entitled to claim the mileage from your home to your full-time job as business mileage (and back). If you start at work, lets assume it is 10 miles from home. You spend 3 hours doing mystery shops after work, driving a total of 40 miles. You can claim 40 - 10 = 30 miles as business miles. If you go home from work, and then start out on your mystery shopping rounds, then all miles from home back to home are business miles. If as you are driving around doing mystery shopping, you drive 3 miles out of your way to your favorite grocery store to do your weekly shopping and you are not mystery shopping, you have to deduct the miles associated with that personal use of your car.
Shopping Southeast Pennsylvania, Delaware above the canal, and South Jersey since 2008