I had him do the dirty work.
He had a professional killer put an end to his boss's life.
I was convicted for having someone kill my b*tchy wife.
You need to have someone shoot him in the head.
I'll have someone talk to him.
Your final sentence is fine in the past tense.
It is possible to use the full infinitive but for different contexts:
- Why do you have a maid?
- I have her to do the cooking and cleaning.
- Why did you have a baby?
- I had him to fill a gaping void in my life.
- Why does he have a dog.
- He has a dog to help him get some exercise.
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