I was doing my PhD for last two years in Basel from a non-EU country. My relationship with my professor wasn't going really well and he fired me when my B permit renewal was going on. I didn't get a a notice period before canceling my contract. Now after 2 months there is no signs of renewal of my B permit.
I am eligible for unemployment pension since I have paid AHV for 2 years
I also went to the RAV and Arbeitlosenkassen but they told me I need to get the residence permit for the unemployment money.

1) Can anyone help me how to get a stay permit to search for new positions.

2) Can anyone suggest me a good Immigration lawyer in Basel to deal with this situation

Normally the RAV will handle the permit, something sounds fishy here.
When did you get given notice & when did you go to the RAV? they should have at the same time.
What does your contract say about notice or was it a fixed term contract.
What passport do you hold.

It was a contract for a year which ended September 30 2011, but my Prof already offered me an extension of 6 months and I applied to the renewal on 29th September, then he told me about the contract won't be extended after 7 days of work on Friday 7 th of October and I went to RAV on Monday 10th of October next working day.
I have proof for me working for these days since I have mails from my professor and from my students as I was a teaching assistant.
I have an Indian Passport.