You can easily add entries for the ATTiny13 to the IDE, but won't show in the same listing as damelis's ATTiny cores, as he hasn't added the ATTiny13 yet.
Here is what I have found for ATTiny13 Arduino core files, working with boards.txt, loading them into the IDE, etc. The boards.txt is all laid out, just drop it into your personal workspace per the directions provided.
How to install
Boards Manager Installation
This installation method requires Arduino IDE version 1.6.4 or greater.
Open the Arduino IDE.
Open the File > Preferences menu item.
Enter the following URL in Additional Boards Manager URLs: https://mcudude.github.io/MicroCore/package_MCUdude_MicroCore_index.json
Open the Tools > Board > Boards Manager... menu item.
Wait for the platform indexes to finish downloading.
Scroll down until you see the MicroCore entry and click on it.
Note: If you are using Arduino IDE 1.6.6 then you may need to close Boards Manager and then reopen it before the MicroCore entry will appear.
After installation is complete close the Boards Manager window.
This will get you quite a lot of functions, configurable clock speeds, etc. The readme for this project helped me quite a lot in understanding what is going on inside the box and how I can enable only the Arduino core functions that I want, to save space.
Microcore is based upon :
And both projects have code updates as of this summer.
For other ATTiny (I'm using an ATTiny85 sometimes since for some things I ran out of space on the ATTiny13)
This will get you:
ATtiny 25 (8 pin)
ATtiny 45 (8 pin)
ATtiny 85 (8 pin)
ATtiny 24 (14 pin)
ATtiny 44 (14 pin)
ATtiny 84 (14 pin)