The phrase "a methodical effort to" is correct and usable in written English.
It can be used when describing a systematic and organized approach to achieving a specific goal or task.
Example: "The team undertook a methodical effort to improve the efficiency of their production process."
Alternatives: "a systematic attempt to" or "a structured endeavor to".
On a second front, after a famously rancorous primary battle with Mr. Bush in 2000, Mr. McCain has made a methodical effort to rebuild his standing with the Bush family.
This phase of the war is what American commanders call a deliberate attack, which means that unless the government suddenly collapses -- an unlikely situation -- the Army and Marine assault will not be a blitz to the outskirts of Baghdad but a methodical effort to destroy, piece by piece, the Republican Guard units defending the capital.
WASHINGTON, Aug. 18 — Senator John McCain is locking up a cast of top-shelf Republican strategists, policy experts, fund-raisers and donors, in a methodical effort to build a 2008 presidential campaign machine, drawing supporters of President Bush despite the sometimes rocky history between the two men.
He will be joined by other volunteers in their methodical effort to persuade voters — often one phone call at a time — that Mr. Romney offers Republicans their best chance to win back the White House.
He mapped, measured elevation, and made the first methodical effort to name mountains.
Mr. McCain's methodical effort to calm conservatives seems to have achieved only limited success, as he was reminded Thursday at a town hall meeting here when a voter challenged him over his stance on immigration.
The great example of political change through legal change was the long, methodical effort to whittle away at segregation from within the legal mainstream that culminated in the court's decision in Brown v. Board of Education.
SCORECARD -- This spot is part of a methodical effort by Mr. Bush's campaign to strip Mr. Kerry of his national-security credentials.
A methodical effort is under way among governors, donors and top Republicans to make the case that a long nominating fight could weaken the party's chances to win the White House, maintain control of the House and gain a majority in the Senate.
Each chronicles Rubin's methodical efforts to inject her life with cheer.
However, no methodical efforts to investigate this have been undertaken.
