H1B Stamped with Client 'A' and travel for Client 'B' with same employer

[Karthik_7000]

Hi All,

I have H1B approved with Client 'A', but they do not have suitable position for travel. But there is another Client 'B' which is only around 50 miles away from the Client 'A', but Client 'B' falls into another state which requires amendment.

Amendment for another employee took 12-15 days on premium processing, but got a RFE and which may take further weeks. 

Now,  Client 'A' is ok to allow me to go for stamping. Since stamping is not employer specific, as i understand, can i travel with that stamped Visa and join the Client 'B' and start the Amendment process immediately after reaching USA.?

For first time travelers, will there be any verification in the port/immigration to provide any specific proof to show that i am going to join the Client 'A'. Like Joining letter, etc? Please let me know.