samsung galaxy s 3 - How do I keep my headphone jack from disabling my bluetooth speakerphone in my car?

I just upgraded to the Samsung Galaxy S III running Android 4.0.4 (Ice Cream Sandwich) and I drive a 2009 Mazda 5 with integrated bluetooth speakerphone.

When I'm in my car, I connect the headphone jack to my car stereo Aux input so that I can play music from my phone through my car speakers. I got a phone call today and I noticed that when I'm connected to the Aux jack, my phone wouldn't connect to the bluetooth speakerphone. In fact, I couldn't hear the phone call at all, although the person on the other end could hear me. When I pulled out the cable from my phone's headphone jack, the bluetooth connection to the speakerphone was restored.

I never had this problem with my Bionic.

How do I make my phone connect to the bluetooth speakerphone even when I have a cable in the headphone receptacle?

Note: my car hooks up to the bluetooth speakerphone only -- it doesn't have an option to play music through the car speakers via bluetooth.

UPDATE (JUL 19)

Having thought about the situation some more, here's what I think is happening:

When I make a phone call using my car's integrated bluetooth speakerphone, my car turns off the stereo input to my speakers and replaces it with the phone input. At the same time, if I have a cable plugged into my headphone jack on my phone, my phone assumes that I want to hear the call through my "headphones" and not any attached bluetooth device, so my phone doesn't transmit the incoming phone audio to the bluetooth antenna -- it sends it to the audio cable plugged into the headphone jack. Since my car stereo isn't accepting auxiliary input during a phone call, I hear nothing.

So, I really need a way to tell my phone that when I have a bluetooth speakerphone connected and "headphones" connected, I prefer to hear the call through the bluetooth connection (I don't think it's very likely that I'm going to be able to get Mazda to change their stereo firmware :-) ).

UPDATE (JUL 24)

1. I tried my wife's Razr (running ICS) and didn't have this problem, so it seems to be Samsung Galaxy SIII problem, not an ICS problem.


2. I've come up with a couple of work-arounds that I'll submit as a separate answer.

Answer

Try an app named SoundAbout. It can let you configure this to work.