If your new employer’s place of business has the legal “no weapons” signage posted, then you had best not consider concealed carry on those premises. IF posted, and you ignore that, then you will likely be 1) fired, 2) in violation of Texas law, and 3) lose your CHL.
If NOT posted, then I would be every careful about concealed carry. Even if this is not addressed in the employee handbook, you can be terminated by your employer for any minimal reason.
Weapons are not permitted in my workplace, but they are silent about concealed weapons in employee’s cars in the parking lot.
Iff’n (another Texas word) I was you, I’d be very careful about what I did at work, and who I talked to about what, until you had established a better understanding of your employer’s philosophies and attitudes about concealed carry.