Long version (skip below for tl;dr)

I recently purchased a new house. (first owner) 3 floors, unfinished basement where the hot water heater is located. Approx 3300 sq ft.

When looking through the house, I was shown a tankless water heater. Red flag immediately goes up. Whatever. I later discovered it's a GAS tankless water heater. I didn't even know that was a thing, but seems like it's the worst of both worlds. My worst fears were confirmed: It takes OVER 2 minutes to get hot water anywhere in the house.

The tank is on max heat (120°F) We have PEX throughout the house, and it doesn't seem like there are any inefficient lines, so my assumption is it must be the water heater itself. Of course a tankless heater takes longer to get the hot water than a standard one, but 2 minutes seems quite excessive.

The "unlimited" hot water is nice once it gets hot, but there must be a better solution.

TL;DR

Gas powered tankless water heater takes too long to heat up, looking for a solution that still allows us to have the benefits of not running out of hot water but also provides hot water in a reasonably short amount of time.

My idea:

Get a small tank *(I'm thinking 4 gal is big enough, but the smallest that is commonly available should be sufficient)* gas powered water heater that I can use to either A) feed into the tankless water heater or B) feed FROM the tankless water heater, but either way basically install the 2 inline so that we have both ready-to-use hot water and also limitless hot water.

My question:

Would it cause any problems with either machine to have hot water fed INTO the machine rather than cold water? Is there a better way to handle this?