Do one's damnedest and Go all out

Go all out and do one's damnedest are semantically related. In some cases you can use "Go all out" instead a verb phrase "Do one's damnedest".

Go all out verb - Perform a task as well as possible.

Do one's damnedest and go all out are semantically related. You can use "Do one's damnedest" instead a verb "Go all out".