Given the power of judicial review, supreme court justices are able to interpret the constitution and declare laws constitutional or unconstitutional. A conservative supreme court justice might be tempted to use this power to declare all laws that do not conform to his particular conservative ideal of the world as unconstitutional. He might see a strong executive trying to make imprudent, radical changes and try to stop that executive by declaring his actions unconstitutional. This is, however, a short-sighted view of the US political system. If each office-holder just wielded naked power in whatever ways he could get away with it would cause lots of problems. Progressives might be okay with achieving progress by any means necessary, but the dispositional conservatives described by Oakeshott are wary of radical actions that don't conform with historical precedent.
Conservative justices prefer the current law of the land and sticking to their defined powers and existing constitutional structures. Conservative constitutional interpretation requires sticking to the tried and true founding document. They like textualism, limiting constitutional interpretation to the text of the document as it would be interpreted by those writing it. Conservative justices like strict-constructionism, only granting powers to government which were clearly enumerated in the constitution. Deviating from the constitution, by giving more power to the judges to enforce their conservatism, could actually be incredibly radical. It could undermine the judiciary itself and probably result in more instability and chaos than the stability they were originally trying to create. Conservative justices definitely channel Oakeshott when they reveal their preference for "the limited to the unbounded" in their use of their own power.
That being said, political appointees by the Republican party may or may not be conservatives as per philosophers like Michael Oakeshott, Edmund Burke, or Russel Kirk. The Republican party must, before all questions of philosophy, perpetuate itself by appealing to a majority of the America people. It's certainly reasonable to critique conservative political appointees as engaging in judicial activism for the sake of those who appointed them. This might not be conservative as per Oakeshott's definition, but might be some form of pragmatic political moving to preserve existing societal structures. Conservatism isn't a clear-cut ideology where there is always a correct answer. Edmund Burke, often seen as the father of modern conservatism said "Prudence is not only the first in rank of the virtues political and moral, but she is the director, the regulator, the standard of them all." A conservative always looking to take prudent action in pursuit of good, not dogmatic adherence to metaphysical or ideological principles.