You’re right, search results don’t stay in the index very long. If there are tweets that you want from the past, it’s best to collect them in real time rather than relying on the Search API…
If you want to keep using the Search API, consider caching your results and using your cache instead, provided you also verify over time that the tweets still exist.
You may be better served using the user timeline API instead and just discarding the results that don’t match the hashtag you’re looking for.