Great questions!
Interesting group the Fremont Hussars. They have their own march.
See https://www.loc.gov/resource/music.muscivilwar-200000471/?st=gallery
NPS has the following:
UNION MISSOURI VOLUNTEERS
4th Regiment, Missouri Cavalry
OVERVIEW:
Organized February, 1862, by consolidation of Fremont Hussars and three Companies Hollan Horse. Attached to 3rd Brigade, Army of Southwest Missouri, Dept. of Missouri, to March, 1862. Cavalry, 2nd Division, Army of Southwest Missouri, Dept. of Missouri, to May, 1862. Cavalry 3rd Division, Army of Southwest Missouri, to July, 1862. District of Eastern Arkansas, Dept. of Missouri, to October, 1862. Cavalry Brigade, Army of Southeast Missouri, Dept. of Missouri, till April, 1863. District of Columbus, Ky., 6th Division, 16th Army Corps, Dept. of Tennessee, to December, 1863. Waring's Detached Brigade, Dist. of Columbus, to January, 1864. 1st Brigade, 1st Cavalry Division, 16th Army Corps, to June, 1864. 1st Brigade, Cavalry Division, Sturgis' Expedition, June, 1864. 1st Brigade, 2nd Cavalry Division, Dist. of West Tennessee, to December, 1864. 1st Brigade, Cavalry Division, District of West Tennessee, to June, 1865. 2nd Brigade, 2nd Division Cavalry Corps, Military Division West Mississippi, and Dept. of Texas to November, 1865.
Thus the 4th MO cav is the successor to the Hussars and the Hussars pre date Feb 1862.
Again from NPS
Fremont's Body Guard, Missouri Cavalry
OVERVIEW:
Organized at Cincinnati, Ohio, and St. Louis, Mo., August, 1861. Attached to Headquarters Western Department, and participated in Fremont's Campaign in Missouri and against Springfield September to November, 1861. Advance on Springfield October 8-25. Wet Glaze October 13. Linn Creek October 14 and 16. Zagony's charge at Springfield October 25, against a force of 2,000 rebels, routing them, and killing and capturing almost as many as the Battalion numbered. Mustered out by order of the Secretary of War November 30, 1861.
Battalion lost during service 16 Enlisted men killed and mortally wounded and 2 Enlisted men by disease. Total 18.
A contemporary journal of Fremont's 100 day command was published in the Atlantic in 1862 and is here. Clearly the Hussars would have been involved with this movement. Remember Fremont took over Union Volunteer forces in Missouri after Nathaniel Lyon was killed at Wilson's Creek.
https://cdn.theatlantic.com/media/archives/1862/01/9-51/132120993.pdf
This contemporary article from the New York Times makes it clear the Hussars were not the same as Fremont's Bodyguard though they may very well have essentially moved and operated cooperatively as basically one unit.
"The best regiment of cavalry will undoubtedly be the Fremont Hussars, now organizing under Major WARING, the officers of which are all appointed by Gen. FREMONT, and only selected from those who have seen service.
But the finest body of cavalry is FREMONT's Body Guard, which is sure to be associated with his renown in the history of the present campaign. The Body Guard consists of 150 young men, as much as possible of the same size and age, the flower of the city. The horses are all of the same color, dark chesnut, and of equal height and appearance. The uniform of the Body Guard, which has been chiefly furnished by the General himself, consists of black jackets and pants, with yellow stripes, of the finest woolen cloth. Capt. ZAGONYI, who is commander of the Body Guard, is a Hungarian, and was in former years commander of Gen. BEM's Body Guard, with which he went out into the wars, 250 strong and bold fellows, of whom only 17 returned with him. Capt. ZAGONYI is a fine-looking soldier, and regarded with the deepest affection by all the men under his command."
See https://www.nytimes.com/1861/09/03/archives/from-missouri-competent-and-experienced-officers-the-attraction-of.html
So in short Fremont's Hussars were a separate cavalry unit under the command of Major Waring organized after Wilson's Creek in St. Louis and disbanded / reassigned In Jan/Feb 1862 to the 4th MO Cav (Union Volunteers).
If you give me the name of your relative, I can dig further for more specific information.
JJR