if you are on that threshold window, why not a period where the stale period is okay? T0-60 seconds, use the first query (don't retrigger a query) T60-120 seconds, use the first query but trigger a single DB query and use the new result. repeat until the key is stale for 600 seconds.
that is, a minimum of 2 queries (the first preemptive one at 60 seconds, (in the cache for 10 minutes total)
and a maximum of 11 queries (over 10 minutes) (the initial one that entered the key, and if people ask for it once a minute, a preemptive one at the end of those minutes, for 20 minutes total in the cache).