In terms of kinetic interception, the next stage is a specialized vehicle called either a KKV (Kinetic Kill Vehicle) or EKV (Exoatmospheric Kill Vehicle).
To understand this, lets go down basic physics. If you want to go higher, faster, you want less weight and mass. A KKV/EKV is a design in maximum minimalism, it only has what it needs.
The problem of using missiles like the SM-3 is precisely that, its a missile. Its a design with a lot of mass that is useless when it comes to space. The fact that the SM-3 is 300kg heavier than the ASM-135 is already a clear disadvantage because of the extra mass, even though you can put more propellant on a larger missile. Even with extra propellant, you can defeat the advantages of this because you would need a larger rocket motor and casing to contain the propellant and that will all add up to the dead mass once the propellant is all burned up.
As a missile for example, you cannot discard the main booster body like you can do with a multistage carrier rocket. Once the fuel is expended, you're still carrying that body like dead weight. Another is the SM-3, being a SAM, has a radar guiding system. Active radar seekers will work against targets at what you may call at "jet plane" speeds but against something at orbital speeds you would need something something with much shorter frequencies to get the precision it needs. Hence when it comes to dealing with orbital speeds, you want a thermal seeker instead, and the SM-3 carrying a second seeker in the form of form of thermal is the key element to this operation. But nonetheless it still carries the active radar system, plus the batteries for it, and that's dead weight. Another dead weight is the explosive warhead, which is meant for proximity explosion against missiles and aircraft targets but means nothing in the dead of space where there is no atmosphere to create a shock wave. And there is more dead weight in the form of aerodynamic controls and their systems within, which is all useless in space.
You have a 1300 to 1500kg missile. You subtract the first booster stage and the fuel expended on the main booster, and how much weight you have left?
In contrast an EKV can only weigh as little as 30 to 50kg. It probably only has an infrared seeker, a datalink, and a small thrust vectoring motor to adjust its flight. The multistage rocket will discard its booster stages once by one, and in the last stage, ejects its EKV payload into space, leaving this small tiny thing moving at incredible speeds. EKV research and development has been up to now, mostly in the conceptual and mockup stages. Except for one country.
Prior to the use of the KT-1 booster in the ASAT test, the two previous launches of the KT-1 booster involved microsatellites (>50kg) which fits the weight category of an EKV.
