One question not answered was whether any annotation was made in your passport for the visa cancellation. Curious what their reason was.
Since you filed for the COS and left before it was adjudicated you did abandon the application but you were not in an overstay
status while it was pending.
Therefore your H1B visa would not be subject to cancellation under INA 222g overstay.
You could then use the existing H1B visa stamp to enter on a new petition filed by your new employer as long as it is approved before the visa stamp expires. Therefore, premium processing would be the only way to accomplish because processing time would take you beyond the visa validity date.
Extensions are only when you are in the US so your employer files for "New Employment". It is "New Employment" because you are not extending a current stay on H1B in the US.
That does not mean you do not get the benefit of the remainder time on your 6 year quota number. Since you have about 4 years remainder time your new petition would be valid for the 3 years max given on approval.
I see this is the first time you are using JA and I am eager to make this a positive experience for you. Please ask follow-up questions if my answer is not clear.
Of course, I am sure you understand your positive rating is very important to me as we are not on salary and that is how we receive compensation, you choosing a smiley face, but more importantly, I want you to come back to the site for all your answers from over a 100 categories of experts!
In the future, you may begin your questions with “FOR JUDITH” and I will be your personal immigration expert.