Ideally of course you would like to fill your open nursing position with a permanent nurse or have a staffing plan in place that lets you know in advance when you are going to need temporary staff to supplement your full time staff.
However, if neither of those are the case, then yes most healthcare staffing companies can extend a travel assignment. With enough notice, most of the time, they will be able to provide you with the same travel nurse who worked the first assignment. But keep in mind that that is really up to the nurse more than the company, which is why it is important to provide a great experience for your travel nurses like you do your permanent staff.
If that travel nurse does move onto another a hospital or company, again enough notice is important, but most companies are going to be able to provide you with another RN to take the assignment.
It is always to the advantage of the travel nursing company to extend assignments, it saves them money in housing, insurance and marketing costs to keep travelers working in longer assignments so don’t hesitate to discuss the option with your healthcare staffing company.