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The plan of the trip could be very clear from where to where you are going, regardless if the mode of transportation, walking, riding a bike, or using a public transportation.

It is legal to carry your pistol from home to your range and back home. My understanding of the Texas law transported, concealed carried or even open carry, carry form does not matter as long as you can "prove" that you going from home to the range. For instance, you are member at the range, you have pre-registered for a match there. Or, you called in and reserved a lane for shooting. Something that a witness can testify at your behest.

To be on the safe side, put it a case/bag/range bag/shopping bag and you are Okay.