Will be able to return on my valid H1 visa if I have a J1 visa stamped on my passport too?

In your situation, you would likely be able to implement your plan as long as you are able to have your J-1 visa stamped in India in May. You could then explain your situation to the inspector at Customs and Border Protection and he or she would most likely attempt to accommodate you. But insofar as you will have entered the U. S. under H-1B visa status, you would have to leave the U. S. and reenter under the J-1 in order for you to be in proper status on July 1. Due to the limitations of the Lawyers.com Forums, Alan Lee, Esq.'s (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.