Technical intern changing company

hi! i'm my residence status here in japan is under technical internship and still working to a company until May or June, 2016. but my residence card is until October2016.. my question is am i capable to find or apply to another company? did the company
will give me a working visa. i just want to stay here in japan. Please help!

No.
you are not working, and you are in training.
after you return to your country, then apply a working visa, if you are qualified.

If you successfully found a new company or the current company will hire you as full time employee then you could get a working visa.
If you have nihongo communications skill it would be easy for you.
May i know how you got an internship? Which company?

in documents, yes! i am a trainee.. but in reality Its the same as a regular Japanese worker.
i got my internship from a sending organization in my country. before that i am a regular employee from a japanese company. then the S.O saw my profile to a job hunting website then contacted me if im interested. i passed all the interviews then studied the japanese language for 3 months. about my nihongo, if the native speaks me not to fast, i do understand them.