The double-leash suggestion above is a simple way to avoid fearful or panicked dogs from slipping out a collar and dashing off during transport, vetting or once handed over to the new owner. Making sure the harness is snug and secure is a must! Below is a poster with suggestions about how long it might take a new rescue dog to settle in and become comfortable with the owners and surroundings - while nothing is set in stone, as every animal is different, it may offer you some ideas about the time frame needed to acclimate a new pet.
Basic tips to remember for avoiding escapes; always take preventive measures such as a harness instead of a collar, use a short (4' - 6' leash), NOT a retractable, and never allow the dog to roam loose when going outside, even for a quick potty break. It is a process for any family to acclimate to a new member, and taking the time to build trust and good relationships is one of the best ways to help prevent lost animals. Enjoy your new friend and be safe!