Beware of whatever you put in place makes families who find they are running late in the morning start to think it's better to keep their child off that day. If a child thinks they are going to lose their break once they are late, they may well refuse to come in at all.
Similarly, if you ring, the number will either show up as "school" or "unknown". Who's going to answer the phone knowing they are going to be told off? You have to start from where the families are, which may not be where you want them to be. For some families, getting out the door in the morning is a major achievement.
Call in parents and explain the importance of being on-time, effects on education of persistent lateness. Listen to what they say are the difficulties and work with them on solving them. What is the atmosphere like on the playground before school opens? Would it put some people off? (i.e. it's easier to face the office staff than the mass of other parents).
Run regular, but not predictable, late-sweeps, get staff out to talk to parents as they come in the gate. Talk to the children informally at lunchtime and formally through school council, run assemblies. Work out if there are advantages in being late - easier to park, no waiting in the cold, opportunity for a cosy chat with a friendly face in the office, avoiding the class teacher asking for a "quick word"?
The "unwritten rule" about Breakfast club needs to be scrapped - it's clearly not there to benefit the children. Could you run a parent and child breakfast club, or have a morning activity/ sports club to encourage children to come in early?