SMS Advanced Client push installation consists of three main phases. Loosely, these consist of:

• Pre-installation phase - The SMS Client Configuration Manager connects to the target client and verifies he client's operating system and client information. The client installation source files are copied, and then the Setup process downloads the SMS core components to the client computer.
•  Installation phase -
  The SMS Advanced Client copies the installation files from the site server and then starts Setup.
• Post-installation phase - After the Setup program is complete, the SMS Advanced Client obtains a site assignment, sends the heartbeat discovery information to the SMS site server, and then contacts the default management point to retrieve the SMS Advanced Client policy.

There is now a KB article

925282 How to troubleshoot Advanced Client Push Installation issues in Systems Management Server 2003
http://support.microsoft.com/default.aspx?scid=kb;EN-US;925282

that discusses all three phases, including:

• A brief description of the flow of events for each phase
• Common issues that can occur in each phase and how to troubleshoot and fix them
• Common troubleshooting procedures, such as Using a CCR file to manually trigger client installation
• Management point troubleshooting advice
• Useful log files