Staying in one spot and day tripping will be expensive if you choose tours. Renting a car in Reykjakik the next day works well, there are ones with central locations like Blue Car Rental, but the downtown airport is not really that far out if you choose Hertz or another international chain. Recommend credit card insurance over the expensive insurance offered onsite, and don't fall for the sandblasting insurance pitch.
General Comment: Everyone stops over in Iceland going eastbound to Europe, which is fine until a person realizes that the flights onward to Europe are very early, that the airport is way out of town, and there's a lot of stress involved in early transport and getting up at 3 am. The reverse direction is afternoon arrival and departure, westbound, so works much better. There is a hotel at the airport which may make sense if adding up all the individual shuttle tickets.
Airport Shuttle: Grayline is cheaper than the big one (Flybus I think) by about half and has a departure that meets the US/Canada arrivals at about 7:30am, and does not involve a transfer to a smaller bus at an intermediate stop the way Flybus does but you may have to walk a couple of blocks to reach a downtown hotel.