Why are you worried about "on or about your person" when you are within your vehicle? The Texas Motorist Protection Act (MPA) explicitly allows you to have a handgun in your vehicle.
PC §46.02. UNLAWFUL CARRYING WEAPONS. (a) A person commits an offense if the person intentionally, knowingly, or recklessly carries on or about his or her person a handgun, illegal knife, or club if the person is not:
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(2) inside of or directly en route to a motor vehicle or watercraft that is owned by the person or under the person's control.
There are only 4 things you need to worry about when driving your SUV:
(a-1) A person commits an offense if the person intentionally, knowingly, or recklessly carries on or about his or her person a handgun in a motor vehicle or watercraft that is owned by the person or under the person's control at any time in which:
(1) the handgun is in plain view; or
(2) the person is:
(A) engaged in criminal activity, other than a Class C misdemeanor that is a violation of a law or ordinance regulating traffic or boating;
(B) prohibited by law from possessing a firearm; or
(C) a member of a criminal street gang, as defined by Section 71.01.
Keep it hidden, don't commit crimes, and stay out of a criminal street gang.