diehard, hard-line, staunch
right-wing
political
compassionate
cultural, fiscal, social
Christian, religious (both AmE)

be
become
remain

extremely, fairly, very, etc.
deeply, highly, profoundly, solidly, staunchly
a staunchly conservative nominee

overly
The gloomy forecasts are based on overly conservative projections of growth.

ideologically
Her views are by no means ideologically conservative.

increasingly
largely, predominantly
moderately, relatively
moderately conservative voters

basically, essentially, fundamentally, generally
She takes a basically conservative view of society.
a fundamentally conservative political outlook

inherently
the army's inherently conservative values

notoriously
Banks are notoriously conservative about their dealings with clients.

traditionally
a traditionally conservative profession

culturally, fiscally, morally, politically, religiously (esp. AmE), socially
the culturally conservative world of commerce and industry

Conservative is used with these nouns:
administration, agenda, approach, attitude, backlash, bench, bias, campaign, candidate, cause, challenger, Christian, coalition, commentator, congregation, Congressman, Congresswoman, councillor, credentials, economist, elite, establishment, estimate, faction, forecast, government, heartland, Jew, Judaism, judge, landslide, leader, leadership, manifesto, media, movement, nationalist, newspaper, party, peer, philosophy, point of view, politician, politics, premier, prime minister, propaganda, regime, rival, scholar, senator, slant, stance, supporter, thinker, value, view, viewpoint, vote, wing