If flights to Tallinn are very expensive or badly scheduled, you can always fly to Helsinki-Vantaa in Finland and cross the sea by 2 hour ferry. No additional documents needed, EU firearms passport and invitation to hunt are enough.
Cheapest ferry prices are something like 20 Euro per passenger and 20 Euro for car. I had a quick glance at Helsinki-Vantaa car rentals and e.g. Sixt is offering cars from 50 Euro per day (there was a mention that cross-border rentals should be requested in advance). So in some cases this could be an option, and also enable you to move more freely in Estonia.