The rules for transferring a non-qualified annuity are much the same as for transferring a qualified contract. There is just less paperwork involved. In either case, there is no tax consequence as long as it is done correctly as a transfer and a distribution is not taken out of the old contract.
Contact the carrier of the new contract that you wish to purchase and request new account and transfer paperwork. In most cases, you will be able to download all the forms you need on the company's website. You will not need to contact your current annuity carrier; your new carrier will do this for you.
Complete the paperwork and send it to your new carrier. You will need to complete the 1035 Exchange form, along with your application and any other pertinent forms that your situation may require. You may have to mail a hard copy, but a signed scanned copy may suffice instead. Your carrier should list its document requirements either on its site or on the forms themselves. Your agent or broker should also know them.
Wait at least two weeks for your carrier to process the transfer. It will notify you when your transfer has gone through and give you your new contract number. Contact the carrier you haven't heard from it after two weeks.
Check with your old carrier and make sure that the entire amount of your contract was dispersed to your new carrier. If you receive your proceeds as a check payable to you, then either you or the carrier have made an error and distributed the proceeds. Work with your old carrier to reverse the procedure and transfer the balance instead.