The recent improvement in the nation's unemployment rate is having one nasty side-effect for several hundred thousand jobless Americans: It's going to mean an end to extended unemployment benefits in all but nine states and Puerto Rico.

Because of the latest drop in joblessness, the Labor Department announced it is cutting off payments beginning Saturday to 293,428 persons who have been receiving 13 weeks of extended benefits beyond the basic 26 weeks provided by most states.

Included among those jurisdictions where these benefits will end are the District of Columbia, Maryland and Virginia. The cutbackw will affect an estimated 2,100 persons in the District. Figures for Maryland and Virginia were not immediately available.

At the same time, effective next Tuesday, the department will end a "federal supplemental benefits" program that provided a second 13 weeks of benefits to some 50,000 jobless persons across the nation. That program is expiring and has not been renewed by Congress.

The two actions together mean that in most states and jurisdictions the maximum unemployment benefits a worker may receive effectively will be cut in half - to 26 weeks from the previous 52-week limit before. The maximum in the District of Columbia will be cut to 34 weeks.

The cutback in the first 13 weeks of extended benefits comes because the jobless rate among persons covered by state unemployment insurance across the nation has dropped below the 4.5 per cent "trigger-point" needed under the law to maintain the extra payments.

Labor Department figures show that for the week ended Jan. 7, the insured unemployment rate nationwide was 4.42 percent - down from just over the 4.5 per cent mark in the previous period. Given the usual lags in the cutoff process, benefits will end tomorrow.

The dip marked the first time that the insured unemployment rate had fallen below the 4.5 percent trigger-point since a brief period from late last July to the end of August. Extended benefits were cut off then, but later were restored.

The drop means that states no longer may continue to qualify for extra benefits unless they have unusually high insured unemployment rates - in excess of 5 percent - and have passed legislation agreeing to federal conditions on support of the program.

The 10 jurisdictions that will continue to receive the first 13 weeks of extended benefits include New York, New Jersey, Pennsylvania, Rhode Island, Vermont, Maine, Washington State, Alaska, Hawaii and Puerto Rico. All have rates of 5.3 percent or higher.

The insured unemployment rate in the District now is 3.55 percent. The city reached a high in September, 1975, when the rate peaked at 4.7 percent.

The drop in the insured rate follows, but is not directly linked to, the overall national unemployment rate. That rate plunged in December to 6.4 percent of the nation's work force, from 6.9 percent before.

Linking jobless benefits to the unemployment rate is becoming a trend in government policy. In his budget last week, President Carter proposed pegging the number of public-service jobs to the national unemployment rate.