Radiator/core support question

[77c10]

I hav a 77 C-10 with a 82 core support and front sheet metal. Until recently it had an LT1 from a Camaro in it and I used the Camaro radiator. I got rid of that motor and went with a 1st gen 350.

My question is, does the radiator set directly on the core support on the rubber pads? I tried it last night and the radiator was about two to three inches too short to reach the mounting brackets at the top. The radiator I bought was a Spectra CU730 which shows to be 28.35" tall. I'm not sure if there is a taller radiator made or not. I got the one that it called for from Auto Zone.

Sorry if this is hard to follow, I will take some pictures tonight to better illustrate what I am talking about if needed.

Thanks,
Tim