Yes, you need to complete a new W-4 form every time you have a significant change like changing jobs, or starting a "side gig".
You should account for the side job somehow, but since it's temporary, you could just enter it as "other income". One warning about side gigs, if you get a 1099-NEC instead of a W-2 form, you will pay an extra 15% in self-employment tax on your net profit (you are allowed to deduct certain expenses).
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