Before positioning the antenna it would be a good idea to test it. If it has been outside chances are there is some corrosion in the connection to the antenna. Check the state of the connection visually first. There should also be a dead short between outer and inner conductor. The connection housing may also have a trimming capacitor inside.
If you introduce another join through the wall you have 2 more places to weatherproof. I would recommend taking off the existing connector, feeding the cable through the wall and sealing it inside and out. An exterior connector is just providing another point at which corrosion can occur. The best solution would be a bulkhead feedthrough plate to provide weather proofing.
The connector to use for VHF is the PL259. I have a stock of these. Is there any marking on the cable to say what type it is? It should either be RG-58C/U (50 ohm) which is about 1/4" dia or if we are lucky, low loss RG-213, which is about 1/2". I have different PL-259’s for whichever cable it is.