"that you first try to win a case and only invest the time required to look for these kinds of connections when it proves necessary?"

No way. You look into these connections as soon as possible - it will help shape your defense - and, if the verdict favors your client, you shut up forever. OTOH, if the verdict doesn't favor your client, you go for a retrial and try to invalidate as much of the prosecution's evidence as possible. It's risky, because now everybody knows everybody's hand, but, in face of an unfavorable ruling, this seems to be the way to go. You should prepare some new evidence to clear your client.