You need Captain time. No PICUS, PF, pilot handling, etc. Did you sign for the airplane? Are YOU the boss? If yes it's PIC. If you're not the Captain it's not the PIC for the requirements.
14,000 applicants for major airlines right now. Probably 10,000 "highly qualified". That's 1000 hrs TPIC(turbine PIC) or more. Midpack candidate at job fairs probably has 6,000 hrs and 2,000 hrs TPIC. 3,000 hrs TT, and zero TPIC, is in the bottom 5%(?) of the applicants. Some do get hired at that experience level but they typically have additional areas on their resume that helps.
Getting more FO time on the 737, 777, Axxx, isn't going to get you higher on the list of qualified candidates. Upgrading on turbine equipment will. U.S Part 121 experience helps.
30-40% of the highly qualified will get hired this year. Many of the 'not highly qualified' are upgrading as the top 30-40% get hired. So in a year or two they'll transition to the 'highly qualified' group. In time the qualifications will lower but it probably is at least a couple of years away.
The higher rated the job the greater level of competition(in general). Atlas Air probably isn't, or won't in the future, attracting the candidates going to the major U.S. airlines and UPS and FedEx.
A year or two ago guys might be trying to get a decent flying job but now guys are getting 3, 4, or more job offers. The top jobs are still hard to get but with 3,000 new hires at the major airlines it's generating movement across the U.S. pilot industry.
Your odds of getting hired right now are very low. To the level that I'd almost guarantee it's not happening. The U.S. industry is set up differently than it is overseas. At 31, if a U.S. major is your goal, your best option is to position yourself for 3-7 years down the road. Get on with a regional that's expanding and hope to upgrade in 2-4 years. By then, with the pilot retirements increasing dramatically, your resume without PIC time might have been selected. With a year or two of U.S. Part 121 TPIC time you'll be much better positioned to get hired. Or you might get hired by a smaller U.S. airline, like Spirit(fast upgrades <5 yrs) or JetBlue. But you won't get TPIC time at JB or Spirit which right now is probably the biggest weakness in your resume to get on with one of the U.S. Big 3(4).