Yash, you are right H1 cap is closed for this year. So even if your company is ready to sponser USCIS will not accept any applications before Apr next year.
In order you to work for a company generally you need to have an H1 sponsered by the company. There are other options which varies case by case.
If you have STEM OPT extension available, all these issues can be resolved and you have to apply for it before the current opt expires. If you do not have opt-extension then better talk to an attorney to discuss you case throughly.