In the US, things work a little differently. Sponsorship is a very foreign concept. While airlines do hire some non-US pilots and citizens from other countries, it's unusual. In the US, generally the qualifications for hire are considerably higher...where an airline in the UK will hire a pilot with 300 hours after sponsoring them, in the US the airline will hire the pilot who has five thousand hours, obtained on their own...and it's competitive enough they'll have 70,000 applicants with that kind of experience, too.
Over the last few years the hiring minimums have dropped considerably to where regional airlines in the US have been hiring at laughable minimums...some as low as 300 hours for a regional airline. However, in general it's not uncommon for competitive minimums to be much higher.
In the US, generally once one has obtained all the necessary certification, one will work for one or two years as a flight instructor, then a couple of years doing charter or freight in small piston twins...and eventually move into small turboprop piston twins, etc. One finds various work such as banner towing, traffic watch, towing gliders or skydivers, etc. Eventually as experience grows and opportunities allow, pilots will find jobs doing corporate, some airline, etc. Even the regionals which have been hiring at extremely low numbers have traditionally required at least 1,500 to 2,500 hours to apply...and then pay a wage so low that many are unable to survive on what they get.
I'd never run across sponsorship until I moved to Australia, and met the Qantas cadet program. I wasn't allowed in the program, but found it interesting none the less. Later I ran into similiar programs by other operators such as Cathay and Lufthansa, etc. They're out there, of course, but not in the US. The closest you'll find are companies which offer programs in which you fly right seat on a small regional type airplane such as a Beech 1900. You have to pay them to sit there, and they don't pay you...these companies will work you to death for two hundred or so hours, then kick you loose, keep your money, and you end up back on the street with 200 worthless hours and no money. All those types of companies do, really, is lower the bar for everyone (what other profession has their employees pay to be there?)...and are generally looked upon unfavorably.
Stick with your course presently. The industry overall is seeing a bit of a downturn. Be patient. This comes and goes in cycles. The US is seeing a very big downturn. (Our company minimums are 4,000 hours, but competitive minimums presently are probably more like 8,000 to 12,000 hours, judging from the applicants we've recently hired)...lots of outfits furloughing or going out of business right now. Things will pick up down the line, just give it some time. Work on your certification, find some entry level work (flight instructing is a great place to start), and as opportunities allow, keep an eye out for work. You'll find it.