"A redesign for 2016 has taken the Tucson from a poor to good rating in the small overlap front crash test," the Institute said in a statement on Thursday.
The test is designed to replicate what happens when the front corner of a vehicle collides with another vehicle or an object like a tree or utility pole.
The Top Safety Pick+ award applies to 2016 Tucson SUVs equipped with an optional front-crash prevention system that includes automatic emergency braking.
The 2016 Sonata also has the optional safety system.
Both Tucson and Sonata vehicles equipped with the front-crash prevention system avoided collisions in Institute track tests at 12 mph and 25 mph.
The Institute noted that for 2016, Hyundai made changes to the driver's seatbelt in the Sonata and the front suspension "in hopes of improving the small overlap rating from acceptable to good. Those changes weren't enough and the rating remains acceptable. It earns good ratings in other crashworthiness tests."
NHTSA has not rated the 2016 Tucson.
Edmunds says: The Insurance Institute for Highway Safety is just beginning to post results of crash tests on 2016 models. None of the aforementioned competitors to the 2016 Tucson and Sonata have been tested.