Arundell married secondly, Anne Philipson, third daughter of Miles Philipson of Crook, Westmorland, by whom he had three sons and six daughers. 
1632 Lord Baltimore Connection
In 1632, after the death of his father George, (1579-1632), the first Lord Baltimore, and late loyal friend and Secretary of State, King Charles I renewed the grant originally made to his father, with the proprietorship of Maryland after an earlier unsuccessful colony of Avalon in Newfoundland. 
Anne Arundel County (misspelled), erected 1650 in the Province and later the State of Maryland is named after Calvert's (second Lord Baltimore's) wife, Ann Arundell. She was born in 1615 at Arundel Castle, gave birth to nine children (three survive to adulthood, with three sons and six daughters) before her death at age 34 on 23 July 1649, and was also buried at Tisbury, Wiltshire. 
1639 Death of Sir Thomas
Sir Thomas died testate 7 Nov 1639, his wife having died a couple of years earlier.
Sir Thomas Arundell and Mary Wriothesley
Thomas Arundell, 2nd Baron Arundell of Wardour (c. 1586– 19 May 1643), who succeeded him. 
William Arundell of Horningsham, Wiltshire, who married Mary Browne, eldest daughter of Anthony Maria Browne, 2nd Viscount Montague (d.1629), by Jane Sackville, daughter of Thomas Sackville, 1st Earl of Dorset. Before her marriage to William Arundell she was the wife of William Paulet (d.1621), Lord St. John, eldest son of William Paulet, 4th Marquess of Winchester. 
Elizabeth Mary Arundell (wife of Sir John Philpot). 
Sir Thomas Arundell and Anne Phillipson
Matthew, born 1609, disappears from view by 1620. Believed by Henry Wright Newman to be Matthew Howard. See Matthew Howard-Arundell for details of the legend. This theory, however, has been discredited.
Katherine (wife of Ralph Eure),
Mary (wife of Sir John Somerset) (and namesake for Somerset County in the Province of Maryland, erected 1666),
Ann, , b. 1615, (namesake for Anne Arundel County, erected 1650. Married Cecelius Calvert, Lord Baltimore. Arundell's daughter Ann, by the above Anne Philipson, later married Cecilius Calvert, the second Lord Baltimore, (1605-1675) in 1628 at age 13.
Frances (wife of John Talbot, 10th Earl of Shrewsbury),
In 1933, Harry Wright Newman  became the most prominent promoter of the legend that Matthew Arundell, son of Sir Thomas Arundell, migrated to Virginia and then Maryland under the name Howard, giving the many descendants of the Maryland Howard family a noble ancestry at the heart of English history.
Several historical facts add plausibility to this legend:
There is indeed a Howard family in the ancestry of Thomas Arundell
There was indeed persecution of the Arundells and other Catholics so that adopting a name from one's maternal line might be attractive.
The Howards rose to prominence in Maryland, suggesting that the family might have prominent roots
Anne Arundell, sister of the Thomas who became the subject of the legend, herself married a husband with the strongest of Maryland connections: Cecelius Calvert, second Lord Baltimore. Weaving facts like this into the legend, Newman suggested that Matthew Howard, ostensibly born Arundell, moved from Virginia to Maryland to be closer to his sister, Baroness Baltimore. Newman apparently was unaware that Anne Arundell, Baroness Baltimore, never left England and never saw Maryland.
Another sister, Mary, married Sir John Somerset, for whom Somerset County, Maryland, was named.
Unfortunately for those who wish the legend were fact, no documented facts support it.
There is no evidence that Matthew Arundell travelled to Maryland. While like other settlers of the day, it can be assumed that Matthew Howard's roots were in England, nothing has been found to identity where or from whom in England.
There is no evidence that the Arundell family acknowledged Matthew Howard or his family as kinfolk, especially Anne Arundell, Lady Baltimore, who would have been expected to acknowledge a brother in some fashion.
The Arundell family were strong Catholics which is consistent with their association with the Calvert family. Matthew Howard, however, arrived in Maryland with the migration of Puritans and Quakers expelled from Virginia for their faith, and settled in the portion of Maryland that was settled by Puritans and Quakers.
Howard-232 and Howard-Arundell-2 appear to represent the same person because: Howard-232 was improperly associated with Matthew Howard-203 and should properly be associated with Howard-Arundell-2. Some cleanup will have to yet be done.