Yes.
It can get odd though. For example, I had a JC installed at Ammond and 1V. I jumped to Ammond to buy some skills, but then realized I needed to get some stuff to JLO. I didn't feel like waiting the 24 hours to jump back to 1V to do it, so flew down the pipe and all to JLO to drop the equipment off. Then I hauled some Trit to 1V. So now both the JC from Ammond and my JC in 1V were in the same place. Oddly enough though, the Jump Clone list on the Medical tab listed my Ammond JC at JLO. So I could jump between 1V and JLO. Woot.
Anyway, I ended up trying to fly back up the pipe, got podded in N-RAEL, awoke in Frulegar where my real Clone is, and now have the same two JCs in JLO and 1V. I now need to jump to JLO, destroy the clone there (which you cannot do remotely), fly back out to Ammond or wherever and reinstall it. It's too bad you can't move a JC like you can a regular clone.
It's hard to bend my mind around this, but this is how I think about it...
I currently have JCs installed at 1V and the Republic University School in a system near Konora. My medical clone is in Frulegur.
Basically, what happens is that when you jump clones, your active body and your jump clone swaps location. So, if I jump from JLO to School, I will now have both JCs in 0.0 space. To reinstalll a JC in empire I will either need to grind faction, or jump from empire back to one of my JCs in 0.0. So, if I jump to empire and want to fly things down, I have to realize that I lose my JC in empire until I either get podded or I can run back there and JC back to 0.0.
One wrinkle is that if you can't have 2 JCs in the same station. So, if I were to jump to school from 1V, my JC in 1V is destroyed (replaced by my school one). To get a clone back at school I would need to jump back to 1V. I would then need to install a new JC in 1V to make sure I had a JC in both stations.
So, here is the way I understand the scenarios:
Starting point:
Medical Clone: Frulegur
JC Empire: School
JC 0.0: 1V-L12
If I get podded, I can take a shuttle to school and JC back to 1V-L12 and my status would look like this:
Medical Clone: Frulegur
JC Empire: School (from 1V-L12), the existing one was destroyed
JC 0.0: none
--> I would need to install a new one in 1V-L12 to make sure I had both.
If I need to go train a new skill and I jump from 1V-L12, my status would look like this:
Medical Clone: Frulegur
JC Empire: none
JC 0.0: 1V-L12 (from School), the existing one destroyed
--> now if I jump back to 0.0 after my skill is trained, it would look like this:
Medical Clone: Frulegur
JC Empire: School (from 1V-L12)
JC 0.0: none
--> I would need to install a new one in 1V-L12 to make sure I had both.
Last scenario - if I jumped from JLO to School because I was too lazy to fly back to 1V-L12, it would look like this:
Medical Clone: Frulegur
JC Empire: none
JC 0.0: 1V-L12, and JLO (from School)
--> I could then choose to JC back to either 1V-L12 or JLO which would reinstall my JC in empire.
One thing to remember is where you left your ship...
Hope this helps to clear things up a little. I hope I have not made the confusion worse.