If I want to work at a big company, I would only consider small companies if the big ones don't want me and not if the big ones just want me a month later.
If you want to work at a company offering more responsibility and/or more money, you consider the company that's offered it to you more favourably than the bigger company that says they like you very much and will decide if you're good enough after a week or two. Especially if you suspect that reflects how the respective companies will respond to any future requests you might wish to make if working for them.
If you only want to work at a big company, you probably don't interview with the smaller company in the first place.
If it takes a month to get their shit together to hire you, how long would it take you to get anything else routine done at that company? You need a new machine for your first day at work and they put in a request and you'd probably finally get a machine a month later. That sounds like a nightmare place to work.
If I want to work at a big company, I would only consider small companies if the big ones don't want me and not if the big ones just want me a month later.