So what does one do when their supervisor asks them to give them a lift to pick up their car at the mechanic? Hypothetical of course. :)
It all depends on whether there's any risk of scandal. Could people who know you're married to someone else and yet do not know the purpose for the trip draw some conclusion about it? If the only ones who see you are people who know that you're just dropping him off to get his car, the risk of scandal is minimal. If no one who might see you knows that you're married to someone else, the risk of scandal is minimal. After all, you could just be a couple (married to each other even). If there's a 30-year age difference between the two of you, the risk of scandal is minimal. But let's say I'm a married 35-year-old man riding with a 35-year-old woman and driving past people who know me and my wife, and they don't know why I'm in the car with this woman, i.e., the reason for the trip isn't known or otherwise obvious, the risk of scandal increases, with people wondering "Are they having an affair?" It's just a question of prudence: How likely is it that someone who sees you might conclude that you're having an affair? Also factoring in is how long are you gone? So, even if a woman who worked in my office asked me for a ride, and everyone knew I was just giving her a ride to get her car, it's no big deal ... unless of course I'm gone for 3 hours without explanation. But if I'm back in 15-20 minutes, then no conclusions would be drawn.