You need to know up front whether you need to be able to dynamically add entries to the filter or if your application can tolerate rebuilding the filter entirely whenever the underlying data changes. In the latter case you have more freedom to choose data structures; many of the modern "better than Bloom" filters are more compact but don't support dynamic updates.
[1] https://en.wikipedia.org/wiki/Cuckoo_filter
[2] https://engineering.fb.com/2021/07/09/core-infra/ribbon-filt...
Cuckoo claims 70% of the size of bloom for the same error rate, and the space is logarithmic to the error rate. Looks like about 6.6 bits per record versus 9.56 bits for bloom at 1%. But at .5% error rate a cuckoo is 7.6 bpr. In fact you can get to about a .13% error rate for a cuckoo only a hair larger than the equivalent bloom filter (n^9.567 = 758.5)
My fairly niche use case for these kinds of data structures was hardware firewalls running mostly on SRAM, which needed a sub one-in-a-billion false positive rate.