I don't think that there is an easy solution to your problem. If there were, and it were easy to put into practice, all the other less-famous labs would also put it into practice and the easy solution would no longer work. It's an old problem in economics.
That said, however, I think two parts of a solution (and by no means all of the solution) are the following (I ignore things like grant money, the fact that you are in Europe, etc. because you either can't change them or will already be well aware of them) :
- Be a good person to work with. Treat your colleagues well, treat your students and post-docs well, share your ideas.
- Increase your sales taskforce. You've probably never thought about the problem that way, but if you are the sole sales-person for your lab, then you are behind the eight-ball. You will find it very difficult to compete with the publicity that other, better known places, command. There is nothing shameful about asking people whom you know and who like you to advertise your good points (that's what post-docs do when they as you for a referral!). So, if there are postdoc who have worked with you or who are currently working with you, and with whom you enjoy a good collegiate relationship, mention to them that you are looking to the future. Tell them you'd be very grateful if they would mention your lab to any third person (potential post-doc?) whom they think worth telling.
To see how important, ultimately, it is to work with people who are friendly, supportive, challenging, collegiate, and all-round "nice", you need only look at the number of questions on this forum (see this example, and this) that are phrased along the lines of "HELP! I'm working in a famous lab with lots of money and a famous PI ... but the PI hates me and I hate them"!