I'll let others chime in who've dealt with the issue before on the early cars, but the biggest problem with the '65-'66 cars of which I'm aware is that the opening in the radiator support isn't big enough to allow enough airflow to help the radiator cool.  You can put a bigger radiator in it, but if its just hidden behind the radiator support its not going to do much for cooling. 

There are so many questions we'll need answers to before we even begin to consider a radiator swap.

1.) What year is the 351? 
2.) What style accessory drive is on the 351 (serpentine belt or v-belt)?
3.) What water pump was used on the engine?
4.) What timing cover was used on the engine?
5.) What kind of fan setup are you running (mechanical, electrical, clutch, with or without a fan shroud)?
6.) When you say it runs hot, what do you mean?  What are you using to gauge the engine temperature (stock gauge, aftermarket gauge, IR thermometer)?
7.) Does the car have AC?
8.) What kind of transmission is in the car?

Keeping the engine cool in these older cars is sort of an it takes a village approach.  It's not something throwing parts at will typically fix, and that's a very expensive way to try and fix it.  The key to cool running is more a case of ensuring that all the parts you use work together and complement one another, rather than thinking one new part is going to solve the issue. 

Here's a link to a write-up that MustangSteve did a few years back....
https://www.mustangsteve.com/radiatorcutdetail.html
 

I cool the 351W in my 68 with the stock opening in the radiator support, but I had a 3 row radiator made to the same dimensions as my stock 2 row 68 radiator. No cooling issues.

MS wrote:

I totally disagree about the hole in the core support being too small for adequate air flow.  My 2017 Shelby actually has air flow BLOCKED OFF to approximately a 6" x 24" opening, and it still cools over 500 HP.  Yes, opening it up might add some air, but a proper radiator, fan, shroud and pulley ratios will cool the 351W.

Before I can answer specific questions, and without me writing a book worth of questions, how about you tell me what you have for pulleys and water pump and we can go from there.  There are two different water pump configurations, right port (1969), or left port (70 and on)

Ah, but starting in the Fox era the Mustang started using radiator airflow designed in a wind tunnel.  Consider how tiny that opening is on a Fox car.  Prior to that it was just use a big honking hole in the front of the car.  Air certainly moved through that hole, but at what static?  What was the pressure drop through the core?  The opening size when you design the airflow using aerodynamics is going to be of quite different quality, and quality wins over quantity. 

New cars also make use of a lot of technology our dinosaurs just don't have.  Engine temperature as a last resort can be controlled by the computer adjusting A/F ratio and timing.  There's also a dual electric fan system that's computer controlled.  Automatic equipped cars have sensors and often computer controlled valves to ensure the trans is at proper temp without getting too hot. 

Not saying the stock radiator opening in the '65-'66 is or isn't too small.  It just gets tricky trying to compare vehicles built 50 years apart.  This is why that information is so important to allowing us to make informed responses.  Trying to solve a car problem over the internet is a bit like trying to give someone a haircut over the phone.  We need to try and get to where we're trying to do it over Skype
 

I have a 351W in my 65, it worked quite even before I installed a Vintage A/C system and it still works quite well.
Cooling System Details …
Aluminum Radiator: Summit SUM-380461
Flow Kool Water Pump (#1680)
Electric Fan: Spal 16” single-speed #30102120
No Fan shroud.
DCC Fan Controller #FK45
180° Thermostat
 

There is a bunch of ideas and theories on how to keep these cars cool. And I am no expert on the subject. This is what works for me . I opened up the support an I am using a 69 radiator that is 24 inches wide. I have dual electric fans from a contour. To control them I made a low/hi control that is adjustable
This is working for me and when I was in Yuma AZ I didn't have any issues with overheating.  Like some others stated timing and fuel ratios do come into play .