I would, your cover letter should tell about you, (be careful not to mention names or specific job desc.) ..."Offer it to them" , before you depart their company, ask if they would like to attach it to your resume so they will have more info about you.
I feel that it is like a letter with a picture attached - it tells a better story about who wrote the letter than the letter by itself...

The way I look at it, it can't hurt to be "over-prepared", if that's possible. You going against a myriad of other candidates for the position, many of whom probably never even gave a cover letter a thought. Always put your best forward at any interview. Good luck!

Matt, I'd play it by ear to see what develops. Definitely have a job-specific cover letter "in your pocket" but I would not commit to it (i.e. staple it to your resume) until I had a chance to view the setting. In your Q you said "take a cover letter." Yes, but I'd wait to hear what the company is doing that day, and exactly how they're doing it. Good luck, Zack

No harm in bringing it. However, my advice is for you to be prepared to deliver what is in your cover letter in person via a clear, professional, thoughtful, deliberate and enthusiastic "voice." A piece of paper will never do the justice of letting someone know who you really are. Good luck.

yes but keep it general. I made a mistake taking a cover letter for a specific position to a hiring event and found another job opening that I was more interested in but couldn't use the letter. Keep it general and usable for any job opening.

Hi Matt: I definitely encourage you to do it! Remember, your cover letter acts as an opportunity to highlight any special skills or experience you have that your resume doesn't reflect. It also provides you with the opportunity to show why you're excited about a job role at a supermarket. It certainly won't hurt your job search, so I'd definitely give it a go!