Thanks Dog5000.

I switched from iPhone as well, and I
actually always did need to adjust the volume louder whenever plugged into the aux cable in the vehicle
with my iPhone.
But with the S3, the first time I plugged the aux in, there was nothing, no sound. Then I went to Settings > Accessory > Audio output mode...where I checked the box, and the music played great?and even at the same volume setting?as the radio was set to (unlike what you experienced with your S3, having to bump the volume up, and like I used to always have to do with my iPhones). But . . .that was only
the first time I plugged the aux cable in. Ever since, even though the
Audio output mode has remained checked . . . I can't get anything to work; no music will play over the vehicles speakers in aux mode! Really strange.
I'm sure the aux cable isn't bad, as I've plugged both an iPhone and a Galaxy Note into the same vehicles aux cable several times, to check the cable, and both the other phones work fine. So it's like my S3's
Audio output mode stopped working ? after the first time I plugged in and put a check in the box ? which did make it work. All I did after that, was unplug the aux cable approx. an hour or so later, after listening to music whilst driving. I wonder if pulling the aux cable out of the S3 ? after that first time it worked ? could have messed something up in the S3? It shouldn't have. I don't think there's any kind of (unmount command) for an aux cable, is there?
Again, really weird, and frustrating.

With the iPhone, for the last three years, never had a single problem. Except for having to boost the volume as aforementioned.