Hilaria Baldwin publicly showed her support for husband Alec Baldwin amid the ongoing investigation into the fatal on-set shooting
Alec and Hilaria Baldwin
Alec and Hilaria Baldwin
| Credit: Steven Ferdman/Getty

Alec Baldwin continues to lean on wife Hilaria Baldwin following the fatal shooting on the set of his film Rust.

On Saturday, the mom of six, 37, showed support for her husband on Instagram following the actor's latest comments about the death of cinematographer Halyna Hutchins. Hilaria posted an intimate photo of her hand on top of Alec's and wrote in the caption, "I love you and I'm here."

Earlier in the day, the three-time Emmy winner, 63, spoke to paparazzi in Vermont, where their family is currently staying in the wake of the tragedy. Although he made it clear that he could not comment on the open investigation, the actor said some words about Halyna, 42, in a video shared by TMZ.

"She was my friend, she was my friend," Alec said. "The day I arrived in Santa Fe to start shooting, I took her to dinner with Joel [Souza], the director. We were a very, very well-oiled crew shooting a film together, and then this horrible event happened."

He also said he has been in contact with Halyna's husband, Matthew Hutchins, and their 9-year-old son, Andros, after they were seen having breakfast together last weekend. "We are in constant contact with him because we're very worried about his family and his kid," Alec added.

Saturday's comments marked the first time the Rust star and producer has publicly addressed the tragic incident in public since releasing a statement on Twitter last week.

"There are no words to convey my shock and sadness regarding the tragic accident that took the life of Halyna Hutchins, a wife, mother and deeply admired colleague of ours," Alec wrote at the time. "I'm fully cooperating with the police investigation to address how this tragedy occurred and I am in touch with her husband, offering my support to him and his family. My heart is broken for her husband, their son, and all who knew and loved Halyna." 

Halyna Hutchins Husband Matthew Honors His Late Wife In First Instagram Post Since Her Death
Credit: Matthew Hutchins/Instagram

Never miss a story — sign up for PEOPLE's free weekly newsletter to get the biggest news of the week delivered to your inbox every Friday.

After PEOPLE confirmed that one crew member died and another was critically injured last Thursday on the New Mexico set of the Western film, the Santa Fe Sheriff's Department said in a statement that Halyna and Souza, 48, were "shot when a prop firearm was discharged by Alec Baldwin, producer and actor."

Authorities responded to a 911 call from the Rust set at Bonanza Creek Ranch at around 1:50 p.m. local time. Halyna was airlifted to the University of New Mexico Hospital where she died of her injuries. Souza was released last Friday from Christus St. Vincent Regional Medical Center where he underwent treatment for his injuries.

Hilaria broke her silence on the incident on Monday by expressing her condolences to the late cinematographer's family in a post shared to Instagram.

"My heart is with Halyna. Her husband. Her son. Their family and loved ones. And my Alec," she wrote. "It's said, 'There are no words' because it's impossible to express the shock and heartache of such a tragic accident. Heartbreak. Loss. Support."

Hilaria Baldwin and Alec Baldwin
Hilaria and Alec Baldwin
| Credit: Jamie McCarthy/Getty

A source told PEOPLE on Thursday that Alec has "found comfort" with family in the aftermath of the on-set shooting. He and Hilaria are currently "laying low" with their six children: daughters Carmen Gabriela, 8, and María Lucía Victoria, 7 months, and sons Rafael Thomas, 6, Leonardo Ángel Charles, 5, Romeo Alejandro David, 3, and Eduardo "Edu" Pao Lucas, 13 months. 

"Alec and Hilaria left the city and are laying low with the kids," the source said. "They do go out occasionally to pick up dinner, but are mostly trying to protect the kids' privacy and just spend quiet time together as a family."

The source added, "Alec continues to grieve the last week's tragic events, but has found comfort being surrounded by his loved ones. He is trying to focus his attention as best he can on being present for his wife and kids as he works through the trauma. Alec continues to cooperate with the sheriff's office in their investigation."

RELATED VIDEO: Sheriff Says Alec Baldwin's Been 'Extremely Cooperative' During the Rust On-Set Investigation

A separate source also previously told PEOPLE, "It's such an unthinkable tragedy and the grief and trauma for him are unbearable. Alec is still in shock and his heart is completely broken. He is doing his best to cope and is leaning on Hilaria and the kids for support."

"Obviously, what happened here is devastating for everyone," the source added. "Alec will need time and space to work through and process this tragedy."