Any time that a pilot flying such a plane with such a configuration (elevator and all moving horizontal stabilizer) senses that they are at risk of running out of pitch control authority they should get on the pitch trim in that direction first and ask questions later.
Keeping in mind that if the pitch attitude is rapidly increasing, it may be better to rapidly roll the aircraft to the nearest horizon which allows the nose to drop while simultaneously using appropriate stab trim. This advice is clearly explained in the 737 FCTM. What has always concerned me is the majority of 737 pilots I have flown with during simulator training, never had this manoeuvre demonstrated during their type rating or recurrent simulator training. They are told by their instructors to "read your FCTM" as if that is sufficient advice to ensure hands-on competence. Well, it isn't.