Adding an additional "order by" column gives me a much worse plan
Your question is missing a lot of detail but I can reproduce something similar.

Setup

CREATE TABLE T1(X INT PRIMARY KEY, Y INT INDEX IX)

CREATE TABLE T2(X INT, Y INT , PRIMARY KEY(X, Y))

INSERT INTO T2 
OUTPUT INSERTED.* INTO T1
SELECT TOP 1000000 ROW_NUMBER() OVER (ORDER BY @@SPID), ROW_NUMBER() OVER (ORDER BY @@SPID)
FROM sys.all_objects o1, sys.all_objects o2;


Query 1

SELECT TOP 1 T1.Y AS T1Y, T2.Y AS T2Y
FROM T1 JOIN T2 ON T1.X = T2.X 
ORDER BY T1.Y;




Query 2

SELECT TOP 1 T1.Y AS T1Y, T2.Y AS T2Y
FROM T1 JOIN T2 ON T1.X = T2.X 
ORDER BY T1.Y, T2.Y




Query 3

WITH T  AS
(
SELECT TOP 1 WITH TIES T1.Y AS T1Y, T2.Y AS T2Y
FROM T1 JOIN T2 ON T1.X = T2.X 
ORDER BY T1.Y
)
SELECT TOP 1 *
FROM T 
ORDER BY T2Y




Query 1 just picks off the TOP 1 from the index in the desired sort order and does the needed joins on the other table for that row. If the join is successful it stops there otherwise it tries the next one in index order until it finds a row that matches or runs out of rows.

Query 2 When adding the new sort column this plan is no longer valid as there could be multiple matches tied with the TOP 1 value and SQL Server decides to join the whole lot and then get the TOP 1 from that.

Query 3 This encourages SQL Server to stick with the first strategy and then just does a TOP 1 Sort on any rows tied with the same value for the first sort key.

For my example data Query 3 works out better than Query 2 but if you have many duplicates tied for the value of the first sort key your milage may differ.

You can try this rewrite and see how it fares

WITH T
     AS (SELECT TOP 1 WITH TIES so.OrgType,
                                ch.Status,
                                rcs.DBSstatusDescription,
                                cid.ApplicationId,
                                ch.DateAdded AS chDateAdded
         FROM   tbl_application_crb_initialData cid
                INNER JOIN tbl_season_organisationId so
                        ON cid.OrganisationId = so.OrgId
                LEFT JOIN tbl_crbHistory ch
                       ON cid.ClientReference = ch.ClientReference
                LEFT JOIN ref_crbStatus rcs
                       ON ch.Status = rcs.statusId
         ORDER  BY cid.DateAdded DESC)
SELECT TOP 1 OrgType,
             Status,
             DBSstatusDescription,
             ApplicationId
FROM   T
ORDER  BY chDateAdded DESC