If your CV, letters, and prior research is good enough to make you competitive for a fellowship, then you could contact some PI's and say you were interested in applying for such and such fellowship and you found they might be a good fit with the work you are interested in and the fellowship. Most people are very happy to have someone come in on a fellowship. Personally, I find it much more appealing just to know that someone is willing to try for a fellowship. It shows a little bit of extra motivation and independence. So even if the person does not get the fellowship I might consider them much more seriously. However, if you apply for a fellowship and your position is contingent on it being awarded, then you should also have a plan B. Ideally, you should work through your network. Spamming people may work eventually, but who knows if what you will end up doing is a good fit (assuming you have some long term vision beyond the postdoc). Replying to ads is good if you find the right one, but there is also a lot of competition in this route, so you really need to match the position well. Again, seeing if your current advisor could reach out to people or you reaching out to people you know yourself may be a much more effective way to get a good position.