Once I have obtained the E1 visa, can I switch to a different company in the US without having to leave the country?

The E-1 [Treaty Trader] visa is issued to persons entering the U.S. solely to carry on substantial trade principally between U.S. and the Foreign State of which s/he is a national. There is no bar on extending your status or transfer to another employer. However, the USCIS will have to authorize the transfer and employment cannot be inconsistent with the terms and conditions of the E-1 visa status.