I'm no historian, so I'm afraid that all I have to offer is based on anecdotal evidence rather than comprehensive research.
In a 1835 book of legal advice to renters and homeowners (F. Locquin, Petit Code des locataires et des locataires et des propriétaires. Paris, 1835.), there are references to moving days (“Des époques de l'emménagement”, p. 48ff.).
Moving times vary from town to town. In most of the country, moving time falls between Saint-Jean [24 June] and Saint-Martin [11 November].
In Paris, one moves in the months of January, April, July or October, unless the lease stipulates otherwise.
One moves on the 8th, at noon, into small dwellings (…), and on the 15th, at noon, into [larger] dwellings, shops, and generally all rentals with a 6-month lease.
While the effective date is the 8th or 15th day of the month, the rent is due from the 1st day of the month. This stipulation causes no prejudice to the tenant since he also benefits from the same delay when moving out.
Another reference (M. Bugnet, Œuvres de Pothier annotées et mises en corrélation avec le code civil et la législation actuelle. Paris 1847) confirms that in Paris, leases start on the first day of January, April, July or October. The text also discusses Orleans, where all house leases start on 24 june in the city and 1 November in the countryside.
There is a reference to a moving week in Delphine de Girardin's Chroniques parisiennes (scanned text, search for “chassé-croisé”). In her letter dated 19 October 1836, she relates moving as one of the major events in Paris in the past week.
Other places in Europe had similar rules, for example I found a reference to the 1734 Swedish property code stating that “in cities, moving day is fixed: in the spring, on the last day of March; in the fall, to the last day of September”. The tradition crossed the Atlantic, where it is remembered in New York City and still current in Québec.
So it appears that by the mid-19th century, moving day was still a thing in Paris, albeit on a three-month recurrence rather than yearly.