Yes I'll enlist down some of the names of some big esteemed companies where you can try for internship to enhance your experience in the same field.

1. Pantaloons
2. Madura Fashions
3. Arvind Brands and fashion designing
4. ITC WILLS lifestyle

About your second question, yes NIFT Delhi does provide you with internships, but they're not into such big companies. Some get the opportunity obviously only the deserving ones do. So try to be the best in your field and then you can apply from outside too.

Best wishes. Thank-you.