Yes, you can simply install in a different directory as the new "named instance" of SQL Server Express.
To install, follow step 8 on Microsoft Install How-To :
On the Instance Name page, select the default instance or named instance for your installation. If you select the default instance, the existing default instance will be updated. If you select Named Instance, specify an instance name
So, what you need to do is specify a named instance and provide your instance name, and connect to it using the URL format as described above.
As Microsoft How-To mentions, the default installation is a named instance as well, with the name "SQLExpress", so if you want to stop or start the service with net start or `pure stop 'you need to write something like:
net start mssql$sqlexpress
and the part of the host name in the connection string for the instance with the default SQL name:
.\SQLEXPRESS (or localhost\SQLEXPRESS)
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