Disinterested Third Party
The original poster simply wants to know if he can skip to the front of the line after primary flight training, so long as he has his four year degree. It's a simple question.
The answer is equally simple. No, you may not, unless you have a tailwheel endorsement.
To get preferential treatment, the degree should be in English Literature, Psychology, or Polar Geology. The college experience should also show a clear and concise record of attendance at political rallies, and involvement in at least one recognized coed team sport that combines rowing and long distance jumping and/or pole vaulting in arid conditions. As a substitute, aquatic equestrian events may be considered, so long as it can be shown that no horses have been drowned, or submerged for periods not to exceed five minutes in tepid water, or ten minutes at subarctic temperatures (Farenheit). Remember, document, document, document.
Because the leap from primary flight training to major airlines is extraordinary, one should be prepared to show broad character development through volunteer activities and community involvement. While any volunteer history of a benevolent nature will be considered, strong preference is given to those with operational experience in the Girl Scouting programs, and documented experience in debris removal along federal or state highway systems, with particular emphasis in roadway segments passing through inner city areas involving a grafiti density of 25.6 hand-painted symbolic references/sq. acre, and should occur between 30 and 60 degrees latitude in the north, and anywhere below 20 degrees latitude in the southern hemisphere. Credit is also given to charity work involving any mammal weighing in excess of 70 kg, or 53 lbs, if notarized documentation can be attached verifying bonding between volunteer and animal, and no external signs of abuse or use with a maze or accordion, as outlined in the International Treaty for the Ethical Application of Mammalian Studies, 2004.
Do not leave this to chance; avail oneself of every opportunity to move directly to a major: choose a flight school that offers flow-through training.
Be aware that some, but not all major airlines give strong preferential consideration to primary flight training students with verifiable musical talent, and may require a recital or demonstration of both ability and aptitude. While most applicants show level 3 (or higher) proficiency with either the banjo or ukulele, successful applicants have also invoked operational knowledge of the marine harmonica, dulcimer, and on at least two occasions involving exceptional talent, the kazoo. Be aware that both were required to operate at a higher level, and at least one said that after the experience, would never attempt it again. Do so at your own peril.
While most applicants for hire direct from primary flight training have been able to show competitive flying skills and superior judgment, the ability to move from primary flight training to a major airline is not predicated on skill or knowledge or judgement alone, but is judged in large part by a single page essay, hand written, double spaced, with 1" margins, explaining why (in the five-paragraph format) one desires to fly for a major airline. Most majors, with the exception of Southwest, employee a committee of three captains and a mid-level flight attendant who volunteer their time three times a month to evaluate these essays and make recommendations to the hiring committee. Southwest has been the sole exception thus far, and retains the older and more formal system of placing essays on a wall and throwing darts. The essay with the most darts gets hired. It's found that blue ink tends to get more darts; your experience may vary.
An older method, though still often successful, is to show up in a pilot uniform and ask to be put on the schedule.