Does a Change in H-1B Employer Require a New Visa Stamp?

Not as long as the current H-1B stamp remains valid. You may simply use the old stamp along with your current-employer I-797 Approval Notice to enter the U.S. (we recommend bringing a recent pay statement from your current employer as well).

The H-1B stamp will need to be replaced only if traveling abroad after the old stamp has expired, though if traveling near the time of expiration obtaining the stamp during that trip may be advisable.