Yes, you may present a new offer if your first offer is not accepted. The thing to bear in mind, though, is that you essentially are negotiating against yourself in such an instance. Working with a capable and informed real estate professional should put you in a good position to formulate your initial offer based on your sense of a property's true market value bearing in mind relevant comps. If your initial offer is rejected by a seller even though it is a reasonable offer, chalk it up to experience and move on. If, however, your initial offer is rejected because it fails to meet a minimum reasonable threshold be prepared to ascend to the reasonable level to successfully negotiate the purchase. As the other comments indicate, once your offer is refused or it expires you are able to reconstruct your new offer in any manner you desire. That includes price, close date, any contingencies, etc.
Best of luck with your house hunting and buying.