Yes, it is safe to vaccinate late in one flu season and early in the next one. The influenza vaccine in March and April will usually be different from the one given in the fall.
The American Academy of Pediatrics (AAP) and the Centers for Disease Control and Prevention (CDC) recommend that seasonal influenza vaccine be administered to children 6 months of age and older as soon as it becomes available.
Because influenza occurs in many areas of the world during April through September, vaccine should be given to travelers who missed vaccination in the preceding fall and winter. Another late season use of vaccine is to complete vaccinations for children 6 months to 8 years who needed 2 doses of vaccine but failed to get their second dose. For each of these situations, vaccine can be given through the month of June since most injectable influenza vaccine has a June 30 expiration date.
Additional Information from HealthyChildren.org: