First, one should note that law enforcement officers obviously try extremely hard to catch murderers. There are no quick and simple solutions which law enforcement agencies are not already thinking about and trying to implement. There are, however, some areas where it might be possible to improve current procedures.
The first area is how information technology is used. Unlike a single murder, serial killings can span a long period and many geographical regions. To investigate them requires seamless access to information systems that are often run by multiple agencies. Cooperation across multiple jurisdictions is essential.
Improvements in information technology use and especially in the ability to create queries that will search a wide range of databases would be useful. Using a statistical technique called "cluster analysis" on victims can be particularly useful as well to identify serial killings.
The key problem in catching serial killers is that many have no intrinsic connections to their victims, making it very difficult to find a starting point for investigation. What is most likely to help is using big data, but this can be problematic as it might involve such things as using facial recognition software to sort through millions of images from traffic cameras or store security cameras near where murders took place, but the level of surveillance necessary to provide such data might also be considered a violation of people's right to privacy, as such surveillance would obviously include millions of images of law abiding citizens.