In Republican-held Texas, claims that the maps discriminated against black and Hispanic voters were ultimately heard, and mostly rejected, by the Supreme Court. In Maryland, Democrats drew maps that left a Republican congressman without a base of support. And in North Carolina, Republican-drawn maps have been the subject of years of litigation and political fighting.

With the justices’ ruling last week, lawmakers were expected to feel more emboldened in their next round of map drawing after 2020.

“If our party doesn’t maintain the State Senate at a minimum, we know the lines will be gerrymandered outrageously by the Democrats and our likelihood to win legislative and congressional seats in this state may be lost for a decade or more to come,” Jennifer Carnahan, chairwoman of the Minnesota Republican Party, said in an email. Minnesota Democrats, who insist that they do not plan to gerrymander, control the governorship and House, and are optimistic about flipping the Senate.

Jessica Post, the executive director of the Democratic Legislative Campaign Committee, which works to elect Democrats to statehouses, said her party was taking a more aggressive approach this cycle, having learned from its string of losses in 2010. Ms. Post said Democrats want to win control of at least one legislative chamber in Georgia, North Carolina and Texas, and to gain full control of state government in Virginia and Minnesota.

Georgia Democrats are outnumbered 105 to 75 in the State House, but say they believe they could win that chamber by focusing on suburban districts outside Atlanta and anti-abortion legislation championed by Republicans. In North Carolina, where the Democratic governor does not have a veto over the new maps, Democrats are looking in 2020 to flip one or both legislative chambers to gain a voice in redistricting.

With 29 state legislatures currently dominated solely by Republicans and 18 by Democrats, many are unlikely to change hands during this cycle, and in those states, the campaigns for 2020 elections have yet to gain steam. (Two states, Nebraska and Alaska, do not easily fit the model.) Still, in some states where Republicans are firmly in control of the legislative chambers and seem unlikely to lose it, Democrats have tried a different tack on the political maps: Pushing to remove mapping power from the politicians and turn it over to nonpartisan redistricting committees, similar to those used in California and Arizona.