There are multiple ways to migrate from JIRA's own custom field to their Custom Field++ versions.
Migration By Modifying JIRA Database
Since Single/Multi Select and Radio Button fields are direct replacement of their JIRA counterparts you can easily upgrade existing custom fields. Easiest way to do it through the database if you can execute SQL statements on JIRA database.
Changing the database directly is dangerous operation, you should backup first and experiment on a test instance.
JIRA stores custom field type information in CUSTOMFIELD table. You need to change two columns in this table, CUSTOMFIELDTYPEKEY, CUSTOMFIELDSEARCHERKEY. The values you should use are as follows:
Migration Using Adaptavist Script Runner Add-On
The other method is using Adaptavist Script Runner add-on. Please refer to Build-in Scripts/Copy Custom Field Values section of Script Runner documentation. You should add new custom field and define the same options as defined in the original custom field.
Migration Using Bulk Edit
If data to be migrated is not huge and there are only a few options. You can also use Bulk Edit feature to move data from one custom field to another. For each option you will execute a search to find all issues having that option. After that click on bulk edit and select the same option for the new custom field. Repeat this process for every possible value of the original custom field. After that you can delete original custom field.