No. But maybe it would be a good idea to avoid any unnecessary hassle from CBP at the airport.
In 2007, USCIS issued a memorandum setting out that so long as the applicant has a facially valid H-1B visa and an I-797 Approval Notice form the new employer, he should be allowed to enter on the visa from the previous employer. (you may Google the Aytes Memorandum, Validity of Certain Nonimmigrant Visas, July 8, 1997, to read it). So as long as you follow these steps, you should be fine.
However, I have also seen persons have trouble on re-entry despite the memo. You have to weigh out the benefits of waiting to get a new stamp at the embassy or relying on the memo.
Unless you think you would have significant "administrative processing" time, the prudent approach would be to get a new visa stamp for the new company. But pursuant to the above memo, from the facts you gave, so long as you have the approval notice and your visa is valid, you should be fine as well. If you choose the latter, it would be smart to have a copy of the memo with you when you de-plane.
Since your company should have an immigration attorney already, have you asked them this question? As you know, this is not legal advice; before you travel you should sit with a knowledgeable immigration attorney, show her or him your documents, and get advice you can rely on before you travel. And it's nice to have their cell number so if you have a problem you may be able to call them.