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You're on the right track but you need to install a return line between the 2 so the water can circulate between the 2 tanks.
Besides the line the pump is on you need a second line connecting the 2 tanks so the water can circulate. Also might help to remove the heat traps from the tank nipples on both tanks. Not that it's relevant to the current issue but is your pump a bronze pump? One other thing, your "crisscross" is not the way to pipe this. The hot out of the second tank should connect to the house not the other tank. Your return should be at the bottom of the tanks. We use the drain tapping for return lines. Remove the tank drains then install a brass nipple and TEE. The branch of the TEE's connect and the drain goes in the end of the TEE. I'd make the solar tank first in line (cold water into the tank and the back up second (hot out of the solar to cold into the back up)