I got a job offer today that I would like to accept. My manager called and left a voice message. He gave quite a few details and said to call him back on his cellphone. But he stuttered and mumbled and it sounded like he said 11 digits, not 10. I tried all combinations I could think of but they were not his number. I can call the number of the branch I have been in contact with. How exactly should I proceed and how exactly should I word this? It’s kind of abrasive to tell my manager “you mumbled when you left me a message” before the first day.
There is no need to make up excuses or blame anyone.
Call the main number of the plant, ask for the manager by name, and eventually talk to him that way. When you talk to him just say "Sorry, I didn't get your cell number from the message.". It happens. There is no need to bring up that you think the reason was that he mumbled. It's irrelevant anyway. If you get his voicemail, just say the same thing in the message.
Remember, he wants to hire you and have you get back to him. He would have liked you to call on his cell phone, but that didn't work out. That really shouldn't be a big deal. There should be nothing wrong with work-related communication via his work phone.
Perhaps he knows he'll be out of the office, and won't get the message for a while. Send him email also. If he has a secretary, leave a message that way too. With all those, he'll find out you tried to get back to him soon enough. He's certainly not going to figure you gave up on the company and offer the job to someone else before checking his voicemail and email.
You are overthinking this.
Email or call the work number of the manager. Failing that, call the company's main number and ask for contact details (or request a call back).
It's not unusual that voicemails are garbled, and people often don't repeat phone numbers when leaving a message (I often have to listen to a VM three or four times before I get the number written down).
I can call the number of the branch I have been in contact with.
You don't have to turn this into a big deal or into some veiled insult.
Just say: Hey Bob, for some reason the voicemail recording cut out for a second when you were saying your number so I had to call through the main switchboard. Do you think you can give me your direct line one more time so I can get it down on paper?
See, it's the voicemail's fault. Darn technology. Oh well.
Tell your manager the truth: he has mumbled. If this manager cares about his self-development he will thank you for the feedback and he will try to be clearer next time over the phone.
edit: If he gets angry because of the criticism then it is not the manager you want to work for anyway.
As for the part how to contact him: simply call the company and explain the situation.