So, MAD is a theory that any country that starts a nuclear war with another nuclear power would take an unacceptable loss to itself, thus rendering Nuclear War a start. What enforces MAD is the existence of a nations "Second Strike" capability by maintaining a "Nuclear Triad".
Second Strike (aka Retaliatory Strike) is the concept that the nation is attacked will have enough of it's nuclear forces survived to hit the enemy nation with a mass retaliatory strike that could cause enough of an unacceptable loss to the aggressor that the First Strike was not worth it. At the height of the cold war, the US military estimated that if they were a victim of a First Strike surprise attack, 97% of all their nuclear forces would be destroyed in the opening salvo, so they built their forces to numbers that made it so a mere 3% of their nuclear forces could inflict these casualties to the USSR.
A Nuclear Triad is a term used to describe the three delivery systems of the war head to the target: Air Based, Sea Based, and Land Based missiles. The USA used stationary silos with solid rocket missiles, which meant that they were ready to fire within minutes but the USSR knew exactly where they were. The USSR used mobile platforms for their rockets with a liquid fuel, which allowed them to move missiles to avoid detection, but the fuel used was so volatile it took them several hours to properly fuel a rocket and every thirty days, they would have to completely drain the fuel or risk premature detonation of the rocket on the launch pad, so it was really obvious when they intended to launch them.
Both forces also used gravity bombs dropped from bombers but since conventional bombings were all the rage during WWII, the defensive measures to deter this system was the easiest to impliment. Still, the US would have constant flights of bombers to the "Fail Safe Point" near the Soviet Border and then return to base 24/7. They also had another portion of their bomber compliment on stand-by, ready to fly with 15 notice warning.
The final member of the Unholy Trinity is the Nuclear Submarine, which is what allowed MAD to truly be successful. Nuclear Subs could stay submerged for longer than the crew could and this allowed both sides a mobile nuclear delivery platform that was somewhere under 70% of the surface of the planet. The survival of these subs and their missiles would be critical to second strike as now a safe first strike was impossible without retaliation. When the actual use of this came into being is not exactly well known, but Nuclear armed subs were fielded by both sides during the Cuban Missile Crisis of October of 1962. In fact, it was recently revealed that one Soviet Sub was discovered by a US destroyer, which proceeded to drop Depth Charges on it to force it to surface. In the Soviet Sub, the Captain had ordered the firing of a nuke on the sub and the First Officer refused to do his duty to launch.
Now, it's important to say this because it's essential to early MAD: At no time was the USSR capable of launching an attack on the United States. However, the USSR had superior intelligence capabilities and would resort to all sorts of tactics to trick the United States into thinking they had the numerical edge. May Day military parades included multiple waves of nuclear capable air planes... which were just the same formation of planes that would fly in circles and change their formation on the next pass. Similarly, the Soviet's ground based mobility meant tracking the number of missiles was very difficult. The CIA estimated in 1957 that the USSR could field 10 prototype missiles by mid 1958... By Mid 1958, this rose to 100 operation missiles by 1960, 500 by 61. Think Tanks in DC, feeling the true intel was classified and worse, gave the number as high as 1,500 at a point where the US operation forces would at best be 130 operation missiles. These number advantages would later be revealed to be gross exagerations... the USSR had a total force including prototype missiles, 4 missiles in their nuclear arsenal.
This is where your change could happen. Had the United States realized early on that this missile gap was in their favor, not to the detriment, they may have been less scared of the demands of the USSR and finally had enough and started a war the Soviets were ill equipped to finish.
As a final note, MAD did not end with McNamara's announcement. The prevailing policy of the Cold War was to ensure that their First Strike did not decimate your ability to launch a second strike. In fact, the threat of SDI in the 80s eliminating the USSR Second Strike was feared by many to be more deadly than Nuclear War. The Soviets panicked so much, they spent themselves into collapse to try and counter it.