H4 to H1 Stamping Required if not going out of the country?

Hi,

June 2014 I came here in USA on H4 VISA. But i applied my H1B VISA on April 2014 through one of the consultant now i have got the approval for the same.

As per my employer i have to go for the VISA stamping as my I94 is for H4.

Could you please suggest me what all option i have if i don’t want to go for Stamping. Can i go for SSN? and start work from OCT1 without Stamping and COS?

Thanks

Vivek Jain