So, basically you are asking can you get admission in another college directly into second year, honestly speaking khushi not all colleges allow direct admission into the second year. It depends on the college  rules. Some colleges allow it if the syllabus and subjects are same in both colleges and if seats are available. So,I think you should send a mail to colleges in which you are interested to take admission and ask them if they accept students in second year.

Along with that you should have some important documents like your first year marksheet ,No objection certificate from your college and a migration certificate.

So,yes there is a possibility of you getting admission in private colleges.

Hope this helps!