I've been a Kroger employee for almost 2 years. I recently moved out and need a second job. I got a call from the Meijer in town while I'm waiting for my Kroger transfer to go though. One of my coworkers said I'm not allowed to work for Meijer? I knew I couldn't work for Wal-Mart but I wasn't sure if the policy was the same for Meijer.
As far as I know, it is against contracts in my division to work for any competitor company (for obvious reasons). Meijer isn't as bad as Walmart, but if I was you I'd stick with Kroger. It's a good "secure" company in the long run.
If they sell anything that kroger sells kroger can claim you are working for a "similar competitor" and fire you. Most companies have these clauses. I've seen people get fired for it, generally they end up telling coworkers and then management gets wind of it. Then they're immediately terminated. I assume your transfer is full time so I would try to hang on. The union told folks to only work fast food (but not subway) or places like fedex. A lot of guys did plasma donation.
