What if I installed a new tankless water heater before the regular hot water heater, so that the tank water heater is getting filled with hot/warm water vs the 40 degrees water its getting now? Our 80 gallon tank turns cold quickly. We had a 40 year old solar unit doing this which has to be removed or replaced, and want to look at all the options.
If your 80 gallon tank turns cold at an unacceptable rate, either you have a broken hot water heater, or you really, really like high water flow. For instance a lo-flo 1.5GPS shower head, given 2 parts hot to 1 part cold, should last 80 minutes.
I would start by looking at the water flow rate out each faucet (i.e. by sticking a gallon bucket under there and timing it with a stopwatch), and comparing that to best practices for efficient homes. Your problem might be right there.
Obviously you should be able to get the lion's share of 80 gallons of water out of the heater before it turns cold. It may be broken. For instance there's supposed to be a pipe or baffle to assure that incoming cold water is deposited in the bottom of the tank. If this has corroded and is dumping cold water at the top of the tank, your outlet pipe will tend to gulp up this cold.
You will not be able to buy another 80 gallon water tank. New tanked water heaters of that size must be "heat pump" types - they use half the energy but make a lot of noise. Also since they are pumping ambient heat into the water, they chill the area around the heater. Your HVAC system will have to work harder in winter, summer will be a wash.
Another complication with tanked heaters is they must now be kept at 140F or hotter to prevent formation of legionella, the bacteria responsible for Legionnaire's Disease. It was always a problem, but now we know it, so now it is a requirement. This higher temperature means more insulation losses and less efficiency. It also will cause scalding, so it absolutely requires new blending spigots with anti-scald features - traditional 2-knob setups cannot be used. This most likely means a low-flow shower valve.
Tankless heaters are excellent if you can provision the large electrical service. They are actually more efficient, since they only heat the water you use, and only when you're using it. Since they don't have an hour to preheat the water, they must do all their heating on-the-spot, so they need more powerful heaters. But you also don't need to pay them to preheat water for an hour or keep it at temperature. Tankless heaters also don't pay for insulation losses. Local heaters can be moved quite close to the point-of-use, so you are not heating a long hot-water-pipe run - nor waiting for it! However to keep power requirements sane, you must keep flow requirements fairly modest. Since they don't store hot water, legionella is not a factor, so you can heat to the more modest and safer (for scalding) 100-110 degrees F.
Using a tankless and a tanked together makes no sense. Putting the tankless after the tanked is useless since the tank must be kept up at 140 to stop legionella, so the tankless would never run until the tank is exhausted. Putting the tankless before the tanked might slow exhaustion of the big water tank, but probably not by enough to matter, and at a huge energy cost.
tank less water heater cost a lot of money to heat water up. on a tank water heater the gas pipe going into it is 1/4 of a inch. on a tank less water heater the gas inlet is 3/4 of a inch. so the cost will come into play. as far as hot feeding a tank. the tank you say cools down quickly. you will still have to use energy to keep the tank heated. only reason you need to do what your asking is so you do not run out of hot water. in this case get rid of the tank heater. or you can use a solar heater in witch you do not pay a gas bill for.the time from cold water to hot water will be shorter.so just to cover everything you have hot water and you drain it. the time it takes to heat the cold water up is two long. buy a better water heater. you could put a tank less in front the cost will double. or use solar. or install 2 or 3 or one big tank water heater