I was in the same situation couple of times before but I have negotiated time instead of severance pay. Negotiate a week notice time if you can. If you tell the new companies attorney they will act immediately. It takes five working days to get LC for your new job and the attorney can overnight your h1b application on same day he receives the LC. The fastest an attorney can do is 7 to 8 days. Technically you are out of status from 6/9 but USCIS tends to overlook 10 days gap. I have seen this with myself, my friend and my attorney told me the same. I also agree with @Shabba regarding pay stubs.
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