In the original short story, Jack notices, when Ennis strips to warsh everything he can reach, that Ennis is not wearing underwear nor socks. That implies that Jack is wearing them.
Oh, in regard to whether Mrs. Twist looked through Jack's left behind belongings, some people have to wait a long time after a loved one, even a child, before they sort through them. For some people, the grief and bereavement process lasts a long time. While Jack's mother never said that her adult son helped her in the house, only that he helped his father on the ranch, I believe that when clean clothing was put in Jack's closet, it was Jack who took it upstairs. Annie Proulx's Mrs. Twist was recovering from an operation when Ennis showed up at her home.
It was more that a year after my late partner/husband, Ed, died before I got any professional help with the loss of Ed and it took few months of weekly sessions to get to where I could be get on with his loss. But, then by that time, I had another big problem I had gotten assaulted from behind and hit in the head several times. Then I had to continue therapy related to that trauma.