WASHINGTON—A conservative watchdog group on Tuesday released 296 pages of emails from former Secretary of State Hillary Clinton’s personal server, including many exchanges that weren’t handed over to the government as part of the Democratic nominee’s archive.

The new emails, released by the group Judicial Watch, offer fresh examples of how top Clinton Foundation officials sought access to the State Department during Mrs. Clinton’s tenure. The documents were obtained through a Freedom of Information Act lawsuit filed by Judicial...