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Please give me the link to the website you read that says you can do a COS and work while waiting to file next April and begin work next Oct 1st?
It is completely false.
"Now, what I heard and read, is that something that can be done is a COS (change of status) that would allow me to start working with the "new sponsor" as early as October 2015 and then have this company applying for a H1B in April 2015 (BTW, my I94 is valid till the end of 2017)."
Often workers change status to a visitor visa which is for a maximum of 6 months before their status expires but they cannot work on that status. They request the COS for time to wrap up their personal affairs and to do sightseeing adventures before returning home. You will not get a year and you need 11 months before your H1B would go into effect 10/16.
If you get a job offer from a university or an affiliate of a university you could be cap exempt and get an H1B immediately.
That is your only option unless you want to become a student and enroll in an advanced degree. Sometimes you can get CPT = curriculum practical training in lieu of going to classes if it is directly related to the degree program and the program director recommends it and the school international office goes along with the recommendation.
Maybe you are confusing this F1 CPT option with a general COS.
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