Hi, this a good question and I would guess as a former owner of a rental home and frequent user of rental units that this is always possible but not easy. I would have rented our home for one night if the price was right and there was no chance of renting it for 4-7 nights around that date.
You will pay for a nights rent and probably for a cleaning fee, so cost may be a factor. Cleaning will probably be at least $125 for a big house.
What I would do to make sure you have booked a cancelable hotel for all of you first. Then, about a month before your trip go to a site like
www.lastminuterentals.net and use the feature that allows you to send a mass email to all owners of 6-7 bedroom homes. Put in the one date you want and then in the "message" section, just tell them what you want and that you are willing to pay a fair price and see if anyone responds.
Second, go to
www.vrbo.com and use the feature where you can check on specific dates. Put in a date 5 days before and 5 days after your date and when the homes come up look to see if maybe there is just a 2-3 day opening between rentals. Someone may be interested in a one night rental if they know they will not get those 3-4 nights between rentals booked.
Third, I suppose, is to get a hold of a company that does rentals and see if they can find a single night. I know from experience that they make money off weekly rentals and searching for one night is not something they want to do. You might find an eager beaver ready to help but do not count on it. And be ready for a huge price if you use a company.
As you know, June is tough because it is a busy season and everyone wants those weekly rentals. With owners you never know. My theory for our 7 bedroom was $300 plus cleaning for one night was better than an empty house!
Good Luck, just make sure you have a back-up plan of a hotel that will let you cancel.
