Not really. The call initiation time is when the recoding options tell it to start. There are several options (IIRC) that control when the actual call recording starts getting dropped onto disk, most of which are buried in the Asterisk code.
Most people (in general) do not want the first 30 seconds (or so) of a call because they don’t want to listen to the system nattering away, or dial tones, or any of the other stuff that happens before the call connects. With that in mind, your request kind of flies in the face of generally accepted practice. IN the systems that I’ve installed, the customer only wants to part after where the customer gets connected to the call. Anything before that is just waste of drive space and noise.