Software Engineer is just a designation in the corporate world. If you have good fundamentals on Computer Science, especially with Data Structures & Algorithms, Operating Systems, Software Engineering and RDBMS(optional but bonus) along with a familiarity in one of the languages (preferably OO based) then you have high chances of getting into big companies from the skillset point of view. Some companies might ask for basic qualifications and some do not even care about it.
Also, you do not need to consider yourself as a Junior developer unless the domain is new to you. That is upto the company to decide your designation anyway. You better focus on showcasing your skillset, rest will be taken care.
One small free advice: Big teams not necessarily equals to huge learning because you'd be a small fish in a big pond so the scope of your work is always limited. I'd rather suggest to join a small startup and with the help of internet resources, you definitely can learn from the best.
Good luck! Cheers!