Actually, the update is one of the usual reasons which may cause data deletion of phone. however. Don't worry too much about it, there are some ways could help you. so had you recovered your deleted text messages now? try to use some data recovery program to assist you to cover your deleted message without backup.
Based on my experience and your information, I would like suggest you to use iPhone Data Recovery from FORecovery to assist you to recover your deleted data from iPhone 6, including contacts, calendars, pictures, videos, documents, etc, which is very useful and practical.
Read the following article about how to use this kind of toolkit for more details if you need.
In addition,what I want to tell you is that it may be a little difficult for you to recover your deleted data if you had used your phone a lot after losing your files because your deleted messages may be overwritten. but I still suggest you to use this toolkit to scan your phone to ensure whether your deleted data exists or not?
Hoping it will work for you and you could get back lost data from iPhone 6Plus after iOS 9.3 update successfully.Good luck to you!!!