How do I set Chocolatey to install applications onto another drive?

Chocolatey FOSS

For each application, you would need to know its command line switch used during installation to change its installation directory and pass it using --installArgs. See Install Command (choco install) and Overriding default install directory or other advanced install concepts.

Another way to ensure a different drive is to relocate your Program Files to a different drive. You may want to look that up; it is possible to do.

Chocolatey Licensed Versions

We've added the ubiquitous install switch! If you need to override the install directory and you don't want to do all of the work to determine what that switch is, you have the option to use one switch with Chocolatey - Ubiquitous Install Directory Option (Licensed Editions Only).

NOTE: We need to ensure the longevity of the Chocolatey community somehow, and that is to have a FOSSium (freemium) model. The Pro version is $8/month (annually $96), costs you less than eating out once a month, gets you some awesome features, and ensures that the community infrastructure continues to provide a great service and improve. While you are using a free service (the community repository, aka https://chocolatey.org/packages), it is not free to provide that service. So we select certain premium features to go into those versions to provide enough value to be worth the price.

I've found another simple trick - install choco as usual, and right after installation move the c:\programdata\chocolatey directory anywhere you like, and then update ChocolateyInstall environment variable and also update PATH environment variable so choco's \bin subfolder is found after moving it.

Of course, I don't know if it fine with any other packages, but I just installed 7zip and docker-machine with no problems, so seems to work.

You could move the Chocolatey directory to another location then create a hard symbolic link from the default location - see The Complete Guide to Creating Symbolic Links (AKA Symlinks) on Windows.

I.e. mklink /J C:/ProgramData/chocolatey D:/my/new/location

But be sure to create the usual backups, restore points, etc. before doing anything.