Altering db schema in production environment
It really depends how big, complicated and active your database is. Bear in mind that while you are making schema changes, many of the tables will be read-only or completely inaccessible until the changes are completed. Also bear in mind that the reverse is true - your schema changes will need to acquire exclusive locks on entire tables and possibly all their relations, which may take a long time or even be impossible to acquire if the application gets in the way. Theoretically this can result in circular locks which will freeze the whole DB. Basically, if there are a lot of reads and writes going on and changes are going to cause significant disruption to your application, then you should schedule downtime for it instead.

Then you have to account for the changes your application will require. Will changing the schema require changes in the application too? Is your application small enough or modular enough that you can make those changes in increments as you change the database? Ultimately, making changes in production is really hard. Unless your app is designed for it from the ground up, it's almost universally not worth the extra effort required for the sake of some small downtime.

The other option to minimise downtime is to build a new database with an improved schema and a new app server connected to it, then build a widget to sync the data between the new database and the old one. When you are ready to deploy you simply direct traffic to the new app server. Again this is a tradeoff between how hard it is to create the system and how hard it is to accept some downtime. There are very few applications that cannot tolerate an hour or two with a "Scheduled Maintenance" message starting 2am on a tuesday night.