Putting additional barriers between critical information and an adversary is a good idea in general, provided it is done effectively.
Obscurity as a meaningful security measure is addressed by the principle of OPSEC (operational security). In short, this entails depriving an adversary of any information which could help them compromise your organization via either social or technical methods.
With good crypto, however, keeping the keys secret is sufficient to protect your data. Any additional technical layers must be justified on their own technical merits.
Cryptography is a very complicated mathematical discipline, and small mistakes can have enormous consequences. When dealing with cryptography, assume the value of an algorithm is virtually zero unless a trustworthy expert indicates otherwise.
In the US, the NSA and NIST are the official assessors of cryptographic algorithms and implementations, respectively. If you don't know where to look for opinions or want a reasonable basis for internal policies, start there.
In practice, wrapping good crypto is pointless. Once an attacker encounters good crypto, they will generally pivot and attack endpoints where the unencrypted data is exposed.
This could be the web server where users supply information, the SCADA system that feeds into your database, or the workstations where employees analyze or monitor the information. If you have a data pump that pulls information from a database and converts it to another format for delivery to a vendor/customer, that's another big target. Alternatively, they could go searching for your keys.
On the balance
Adding features to an application has a non-zero chance of introducing bugs and complicating troubleshooting. This can lead to extended downtime when something goes wrong or slower responses to security vulnerabilities in the application. Due to these factors, a security feature of little-to-no value is a detriment rather than an improvement.
If you wish to protect yourself against a potential attack on AES, then you should simply use another crypto standard based on a vetted algorithm rather than developing your own. You will get better security with less effort.