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Hugh Jackman and Sutton Foster took the next step in their relatively new relationship Wednesday ... they're now Instagram official! Instagram / @thehughjackman

Hugh posted a pic of his new flame getting up in front of a crowd at a jazz club in New York, writing ... "@suttonlenore performing during the holidays at @cafecarlyle ... now that’s an iconic NYC night! And, truly magical."

This comes after the couple went red carpet official last month, holding hands and posing together at the AFI FEST screening of "Song Sung Blue."

The 'Music Man' alums and longtime friends went public with their relationship in January, showing PDA during a dinner date in Santa Monica. Gallery Launch Gallery Getty

As you know, Hugh and ex-wi

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