Any idea as to where the pipes going to the pump come from?
Is your house a single story or multi-story?
I noticed there are two runs coming into the pump, are they coming from different areas of the house?
For example, in the farthest bathroom, is there a return line off of the hot water line?
One thing you can do is to use a gravity type water hot water circulation system and you don't need a pump to do it.
First, in order for it to work, you need a return line from the hot water faucet that you want to get better response from.
Then, you need a check valve to ensure proper flow.
Finally, you need to plumb the return line into either the cold water supply or the the drain valve as you have.
As the hot water rises, it loses it heat and becomes more dense. With no place for it to go, it kind of hangs near the tap as the hot water keeps pushing it up. With a return line, the colder denser water now has a place to go, back to the water heater. The check valve prevents it from going in a reverse direction.
Essentially you have created a convection current or loop. It won't work on a tankless unit though. That would require a pump. It will only work on a tank unit. And only when the water heater is lower than the faucet (tub, shower, dishwasher, etc...) that you want to get hot water to faster.
I had it in one of my old homes and it worked great.
P.S. get rid of those gate valves and put in some ball valves.
And be sure to use sharkbite type connectors!! (JUST KIDDING!!!)
Also, I would suggest getting rid of that galvanized TEE and what looks like PVC, or maybe galvanized pipe coming off of the tee. That is a failure waiting to happen.