Previous company threatening arbitration when new job has nothing to do with previous job

To elaborate on the jobs, the previous job is pest control. I spray outside the residence for weeds, moss treatments, irrigation, and control outside pests. The new job is just a lawn company that cuts grass and does outside labor work on peoples yards. At this new job there will be no spraying of chemicals just regular yard work. I have asked to see a copy of my agreement, but my previous company is refusing to let me see it.

My husband has been working for a company, Florida Pest Control, for over two years. A contract was signed saying he is an at-will employee, and that if he goes to another company that he cannot give out trade secrets, customer info, etc. He put his two weeks notice in two weeks ago with today being his last day. Today, his last day being there, he was threatened by a district manager that he could not work for a company he got hired on, which is a Lawn Agency (cutting grass, etc) that does no pest control work. Can his previous company keep him from working at a company that has nothing to do with pest control telling him that he would end up going through arbitration? Through the last couple of weeks the management has been telling him that he is making a big mistake and is refusing to give him a copy of the contract he signed or any other copies he made need. Is this right?