That verb is appropriate and idiomatic to use in that situation, but the verb tense should be the simple past because the action (of softening) takes place in the past, and the verb in the first sentence ("didn't want") is in the simple past tense.
My boss didn't want me to give a day off because I have already had one this week. But then he softened and gave it to me.
- Edit:
Sorry I didn't notice this when I first copy-pasted your original sentence, but there is also a word order issue. The sentence should be, "My boss didn't want to give me a day off because I have already had one this week. But then he softened and gave it to me."