The human body already has two distributive fluid systems, the cardiovascular system and the lymphatic system. These two are actually connected, the lymphatic system returns extracellular fluid to the blood vessels.

No cell in the human body is more than two cells distant from a blood capillary.

Everywhere that blood is delivered to (except the CNS) also accommodates lymph capillaries.

It sounds like the space around cells is already pretty occupied. Since you want your system to provide something to cells in the same way that blood does (I'm assuming), then proximity is very important. Your system and the cardiovascular system both have to have very short, permeable distances to every cell in the body.

This will be the main difficulty in proving that your new circulatory system can exist. How can it service everywhere it needs to without pushing cells in tissues too far apart? Regarding your titular question, I sadly don't think it's possible to add another capillary system without changing something about the way the body already works.

Fitting in your larger vessels and organs is actually very realistic. The heart has dimensions of about 5 by 3.5 by 2.5 inches. Between one and three in ten people have an extra spleen (supernumeracy, the physical condition of having additional organs or body parts, is extremely common). A spleen is normally about 5 by 3 by 1.5 inches. Although smaller than a heart, this is just one example of supernumeracy I grabbed at random from the first Google result. I think it proves the body's more than squishy enough to accommodate a lot more than you'd expect without noticing.

The human body already has two distributive fluid systems, the cardiovascular system and the lymphatic system. These two are actually connected, the lymphatic system returns extracellular fluid to the blood vessels.

No cell in the human body is more than two cells distant from a blood capillary.

Everywhere that blood is delivered to (except the CNS) also accommodates lymph capillaries.

It sounds like the space around cells is already pretty occupied. Since you want your system to provide something to cells in the same way that blood does (I'm assuming), then proximity is very important. Your system and the cardiovascular system both have to have very short, permeable distances to every cell in the body.

This will be the main difficulty in proving that your new circulatory system can exist. How can it service everywhere it needs to without pushing cells in tissues too far apart? Regarding your titular question, I sadly don't think it's possible to add another capillary system without changing something about the way the body already works*.

Fitting in your larger vessels and organs is actually very realistic. The heart has dimensions of about 5 by 3.5 by 2.5 inches. Between one and three in ten people have an extra spleen (supernumeracy, the physical condition of having additional organs or body parts, is extremely common). A spleen is normally about 5 by 3 by 1.5 inches. Although smaller than a heart, this is just one example of supernumeracy I grabbed at random from the first Google result. I think it proves the body's more than squishy enough to accommodate a lot more than you'd expect without noticing.

* For example, a new blood capillary layout that aligns with yours to optimise cells reached from each system, and/or sparser initial lymphatics, leaving space for your system's capillaries, and relying on superhuman lymphangiogenesis for clearing out extracellular fluid when it matters.