Conway seem to be focussed on Deloitte when I was with them last year.
For the major big-4
Cap is recruiting - apply direct/referral if you can.
Acc is recruiting as you know
EY - might be on hold but worth checking.
If you already have an offer, are you sure you can keep hold of it while you go through the other processes as they all take time?
What do you not like about the current offer/company that makes you want to look around? Sometimes having multiple offers can be a downside as you have to say no to all but one which *could* be taken negatively if not done politely with good rationale.
Looks like you might have an offer from Acc - would imagine that it would be competitive and unless you are looking for anything specific should be as good if not better than others.