Alec Baldwin's niece shared some somber words on behalf of all affected parties, taking to IG to write ... "Sending all my love to the family of Halyna Hutchins. This is a truly unimaginable and devastating tragedy."

She adds, "My thoughts are also with Joel Souza as he recovers. I am absolutely heartbroken for everyone involved." No explicit mention of Alec -- to whom she has a direct tie -- but it's obvious why Hailey's weighing in ... as this is a family affair.

As Hailey sent her own thoughts ... Joel Souza -- the director of the Baldwin flick, "Rust," who was also injured during the accident -- has broken his silence on the tragedy. He tells Deadline, "I am gutted by the loss of my friend and colleague, Halyna. She was kind, vibrant, incredibly talented, fought for every inch and always pushed me to be better."

He adds, "My thoughts are with her family at this most difficult time. I am humbled and grateful by the outpouring of affection we have received from our filmmaking community, the people of Santa Fe, and the hundreds of strangers who have reached out. It will surely aid in my recovery." Souza was reportedly released from the hospital Friday with a shoulder injury.

Finally, Hutchins' widowed hubby, Matt, also shared some thoughts publicly for the first time -- writing, "Thank you to Halyna's friends and mentors at AFI for establishing a scholarship to honor her memory and support aspiring female cinematographers."

He continues, "Anyone seeking to honor her memory, please direct your giving to this fund."

Alec had previously said he was in contact with Matt ... and was cooperating with law enforcement to figure out how exactly this might've happened. There are already some clues, however, that have surfaced in a warrant -- which said an assistant director handed Alec the hot gun, thinking it was "cold."