For the exact example from above the solution is here. I haven't tested it for multiple patients or multiple missing dates. But I can, if it's not enough.

CREATE TABLE Patient_Locations (
    Patient nvarchar(100) NOT NULL
  , Admission_Date date NOT NULL
  , Discharge_Date date NULL
  , Location nvarchar(100)
)

INSERT INTO Patient_Locations (Patient, Admission_Date, Discharge_Date, Location)
VALUES ('ABC',  '20210102', NULL, 'NURSING HOME')
     , ('ABC',  '20210203', '20210204', 'ICU')
     , ('ABC',  '20210410', '20210413', 'HOSPITAL')

;WITH Next_Known_Dates (Patient, Admission_Date, Discharge_Date, Location, NEXT_Admission_Date)
AS (
    SELECT Patient
         , Admission_Date
         , Discharge_Date
         , Location
         , NEXT_Admission_Date = LEAD(Admission_Date, 1) OVER (Partition by Patient ORDER BY Admission_Date)
    FROM Patient_Locations
)
-- First row where Discharge_Date is Missing. Replace by next known period
SELECT Patient
     , Admission_Date
     , Discharge_Date = NEXT_Admission_Date
     , Location
FROM Next_Known_Dates
WHERE Discharge_Date IS NULL
UNION ALL
-- Just normal rows
SELECT Patient
     , Admission_Date
     , Discharge_Date
     , Location
FROM Patient_Locations
WHERE Discharge_Date IS NOT NULL
UNION ALL
--Little magic. Replace missing dates with 'normal' Discharge_date and NEXT_Admission_Date
SELECT Patient_Locations.Patient
     , Next_Known_Dates.Discharge_Date
     , Next_Known_Dates.NEXT_Admission_Date
     , Patient_Locations.Location
FROM Patient_Locations
JOIN Next_Known_Dates 
  ON Next_Known_Dates.Patient = Patient_Locations.Patient
 AND Next_Known_Dates.Admission_Date > Patient_Locations.Admission_Date
WHERE Patient_Locations.Discharge_Date IS NULL
  AND Next_Known_Dates.Discharge_Date  IS NOT NULL
ORDER BY 1, 2, 3

You can try it here: https://sqlize.online/sql/mssql2022/14e363a81956b2e9019fc44daf47a8c7/

Look at this select for the Idea of a new column NEXT_Admission_Date:

SELECT Patient
     , Admission_Date
     , Discharge_Date
     , Location
     , NEXT_Admission_Date = LEAD(Admission_Date, 1) OVER (Partition by Patient ORDER BY Admission_Date)
FROM Patient_Locations