France's largest cities — Paris, Lyon, Marseille, Toulouse, Bordeaux and others — have access to round-the-clock veterinary emergency care, either through a dedicated 24-hour clinic or a hospital-affiliated emergency service that a regular vet will refer you to after hours. Paris in particular has several centres offering genuine overnight and weekend emergency cover.
In smaller towns and rural areas — which make up a large share of France geographically — round-the-clock coverage is less certain, and the regional on-call rota described in the Emergency Vets article becomes the more relevant system. Ask locally as soon as you arrive, since availability can shift seasonally in popular holiday regions.
Given the cost of a genuine emergency, it's worth having a payment method ready that works smoothly for a larger, unplanned charge — some clinics expect payment at the time of treatment rather than billing afterward.