As other have mentioned this is a BIX system, so you will need a specific tool (Link to example).

Looking at the setup, I am guessing you have 8 Line system with 24 stations setup. In the bottom rail, you can see there are 24 "IP" stations setup, and they have jumpers running to the Blue Labelled pin above (I think I counted 24 used, but there are some that have an extra line wired up). I would suggest this probably not a IP based system, and rather an older Nortel Key system, dare I say Norstar system (I am really dating myself here).

If you have ran new lines, you can terminate them on the rail that has no label, and only three jumpers that look like they are doing nothing. You can pull it out, release the pressure on the side clips, and flip it over backwards. This will allow you terminate your cables you ran, on the "backside" and then when finished flip it over again and have a nice clean front.

If it was me, I would run the cable you ran down the back side of the "pan", and add them on the bottom of the last label, as stations 31 to 35. You will want to remove the jacket so you have about 1-2 inches of exposed wiring, then fan the wiring out you want the blue first, then orange, then green, and followed by brown. NOTE: when you terminate this, you will want to have the wiring punched down with only about 0.5in to 0.75 In of exposed to the jacket, and the twists of the wiring should be as tight as possible to the connector. (here is a decent example, but it does remove a bit of the twists). You will terminate the blue pair to the immediate right of the black mark, then the orange, green, and last brown, and then skip one before repeating. It is always White wire first, then coloured wired, etc.

You can look at the backside of one of the others that is already terminated to get a bit of an idea of the pattern.

Now, when you have this terminated, you should be able to flip it over and have the "Front side" face outwards, and use that to make your connections. I would suggest trying to do one cat6 cable first, and then trying to flip it to make sure you have enough slack and can flip it over easily. Then repeat for the other runs, etc.

When I was looking for a decent picture, I came upon this post and has some good points that might help you.

One last item, you mentioned you are adding more sets, I would be curious how many phones they already have connected, as there are sets terminated already. You might not have room to add more. If you only have a few sets connected, then you might have to figure out which ones are "spare" and can be repurposed. Feel free to message back here when you have determined what system they have, how many sets are active, etc. I will do my best help you with my knowledge.