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The general rule is that a private employer, with minor exceptions such as minimum wage, discrimination, and others) can have any work policies they choose to have. Unless an employer has a firm commitment or contract with employees that all jobs will be posted, or that new postings will be available to internal employees before others, or with a particular employee that they will have first option, or other such types of contracts there is normally no requirement for a private employer to post open jobs at all or to post them in any particular way.
The TX Workforce Commission has this to say about job postings here and you should contact them to see all requirements that apply to employers in TX.