It will be on the monthly Chess statements you receive if you buy or sell shares within the month.
You could also log on to the share registry company's web site of any stock you own (most are either with Computershare or Link Marketing Services). I think you can access your shareholding on these sites even if you do not have an account number set up with the registry (it should ask you for a ticker of one of the shares in your holding and then it will ask for some personal data to authenticate you.).
If you know the Broker Account, under which the shares were bought, your Broker will be able to tell you the associated HIN. They will of course request proof of identity.
If you can present a copy of the Contract Note, your chances will be greatly enhanced.
Not sure if there is a channel to move to SRN directly, from memory. You can transfer the company shares to other brokers with a 12A form, if you're with CommSec, from memory.