You have conflicting requirements here. The compatibility requirements forces you to keep the old hashes. The security requirements forces you to drop them. You will have to make a choise here about what requirements to fulfill.

If you decide to keep the backwards compatibility, try making the best out of a bad situation:

• The old hash and the new hash should be stored in different tables, and the database user that the web application uses should not have read access to the old hash. Use table and/or column permissions for this.
• As soon as you no longer need the old application, drop the table with the old hash. Ashley Madison famously failed at this point - they upgraded to bcrypt, and then for some idiotic reason they left the old MD5 hashes lying around in the database. When the database was leaked, that fancy bcrypt did not help much...

Or, alternatively, if you are not afraid to create a bit of a mess:

• Drop the old hashes. In the new application, add on option "Create temporary password for old application". It gives you a long, random password that is hashed in the old way and only kept in the database for X minutes. The user can then logg in to the old application, and the password is then automatically deleted.

You have conflicting requirements here. The compatibility requirements forces you to keep the old hashes. The security requirements forces you to drop them. You will have to make a choice here about what requirements to fulfill.

If you decide to keep the backwards compatibility, try making the best out of a bad situation:

• The old hash and the new hash should be stored in different tables, and the database user that the web application uses should not have read access to the old hash. Use table and/or column permissions for this.
• As soon as you no longer need the old application, drop the table with the old hash. Ashley Madison famously failed at this point - they upgraded to bcrypt, and then for some idiotic reason they left the old MD5 hashes lying around in the database. When the database was leaked, that fancy bcrypt did not help much...

Or, alternatively, if you are not afraid to create a bit of a mess:

• Drop the old hashes. In the new application, add on option "Create temporary password for old application". It gives you a long, random password that is hashed in the old way and only kept in the database for X minutes. The user can then logg in to the old application, and the password is then automatically deleted.