H1 B Extension- Limiting validity notice by USCIS

I’m in 6th year with H1 B visa and also have I-140 approved a year ago. Current Visa or I-797 expired in March. My Employer filed for my H1 B extension but USCIS responded with limiting validity notice stating H1B status allowed for maximum 6 years and they will extend till June only(considering vacation period) . Not sure if I-140 approval notice was not seen or not filed. USCIS mentioned to file the motion if we disagree with this decision along with form I-290 B. If no response in 33 days, this decision will be final. My Employer or their attorney suggested to wait till we receive approval notice and re-file for extension instead of filing the motion which will take too long time. Is it a better idea to wait till my approval notice comes and then file again for extension? Meanwhile, can I switch the job and new employer would file for H1B transfer? Any risks? Even for this do I need to wait for latest/new approval notice?

I suggest to go by what your attorney is planning. They know much more information about your case and can take better decisions.

You should be able to file H1B transfer if your current i94 is valid without any issue.