Well, France- or various French explorers, definitely tried to get an earlier start than that. So I wouldn't say they refrained exactly- they just weren't terribly successful.

The earliest claim I know of comes from the attempt at building a fort/trading center off the coast of Brazil; Fort Saint Alexis in 1531. The fort did not last very long- the Portuguese captured the men and ships involved in its founding.

In 1536 the French claimed Saint Pierre and Miquelon which lay not far from the Eastern Coast of Canada and which represented an important stopping point for European fishermen crossing the Atlantic- though full time settlers did not arrive in until the 17th century.

In 1555 France established France Antarctique, again trying to establish themselves in what is today Rio de Janeiro, Brazil. This is perhaps the largest attempted settlement/claim by France in the 16th century. A nobleman by the name of De Villegagnon lead 600 predominately Hugenot/Calvinist colonists to develop the area. The work was hard and de Villegagnon was not terribly well liked by all the colonists, some of whom gave up and went back to France where they helped dissuade further settlers from heading to the colony. In the end, Portugal pressed its claims to the area and removed the French colony.

In 1562 France established its first settlement on continental North America at Fort Caroline (near modern Jacksonville, Florida). Several hundred French Huguenots settled in the area after French exporter Jean Ribault had thought it would make an idea and well protected settlement. This time the French colony ran afoul of Spanish claims to Florida who destroyed the settlement and killed the Huguenots as heretics.

Other than that, French colonization and claims really take off in the 17th century with Acadia/Nova Scotia in 1605, New France/Quebec 1608, followed by a plethora of Caribbean islands, Indian Ocean island, African coastal islands, and India trading stations.