Use your cold call to get someone interested. Follow up by outlining what you do to design/improve their site. If you are looking for 50% up front before starting the project you may be able to do it if they're interested after that; break up line item a quote for them (hours and hourly rate, hosting and domain if needed, any premium theme or code or plugin that would cost money) and then invoice them for 50% off that.
You are not going to be able to just call someone, ask if they want a website and then ask for some $ saying its a 50% deposit and get their cc all on your first call with them. The cold calling should just be located a potential client not locating them, selling your service and getting paid for it all at the same time.
In fact I would recommend you call and find interested people, then follow up with what your work would be. They c an provide feedback and if they're on board then you give them a quote. After this do work and present it as a preview/example so everything is good, if not you can work on a rate for revisions if they're major. If it's all good then they can pay and you will deliver the site.
It won't be easy to get a deposit upfront before beginning work even but it will be almost impossible to get them to read off their cc # for a deposit on the very first cold call you make to a random business.