Without a court ruling or parole board recommendation, there is only one person that can stop the execution, and that is the governor, who has the unilateral authority to grant a 30-day stay. This power may be used only once per prisoner. Even if the execution ends up being delayed for years, when it is ultimately reset, the governor (or his/her successor) may not issue another stay for that prisoner. Because of this, the governor always waits until every other last resort has failed, which means that if a stay is issued, it will be within hours or even minutes of the execution time.
If the governor is out of the state, the emergency appeal authority rests in the hands of the lieutenant governor. The lieutenant governor is independently elected and may be of a different political party and/or ideology than the governor. The absent governor may express his opinions or wishes, but the lieutenant governor is not legally required to defer to them.
Although the governor's power to intervene is called a 30-day stay, in reality it is usually longer -- months, years, even decades -- before the execution is carried out. But, it cannot be carried out for at least 30 days, and any further delays beyond 30 days are under the control of the courts and the parole board, not the governor.
http://www.txexecutions.org/primer.asp