Here is our list:
Edward Dickinson Baker- US senator from Oregon, not officially a general because he still held his Senate office while commanding his brigade, Lincoln had named him a MG, which to accept, Baker would have to resign his seat. A close friend of Lincoln's. Killed at Ball's Bluff 21 Oct, '61 while inexpertly commanding the US forces in that disaster. His death led Congress to create the Committee on the Conduct of the War, overseeing military affairs, and a great thorn in Lincoln's side throughout the war;
Nathaniel Prentiss Banks- a ranking MG, former Republican Speaker of the House and Massachusetts governor, no previous military experience save militia musters. Led corps, armies, and departments throughout the war, usually to the detriment of US forces in the field, however he did take the last Confederate bastion- Port Hudson, on the Mississippi opening that river in its entirety to the Union. Note, this was after Grant had taken the major bastion of Vicksburg the week before;
Benjamin Franklin Butler- a ranking MG, a prominent War Democrat who had been actively involved in the Massachusetts militia, leading the early Massachusetts contingent to Washington's defense. Secured Baltimore and Annapolis early to the Union. Established escaped slaves as contraband of war while holding Fortress Monroe, took and administered New Orleans, led the Army of the James till removed by Grant after the Ft Fisher fiasco, Jan, '65;
Daniel Butterfield- prominent New York militia leader and businessman, held various division, corps, and staff positions throughout the war. Notably Hooker's chief of staff at Chancellorsville and Meade's at Gettysburg, where he was wounded and subsequently shelved by Meade. A particular crony of Hooker and Sickles, he seems to have been an unlikable and conniving sort who rubbed most of his colleagues the wrong way;
John Charles Fremont- a prominent Republican and abolitionist with strong ties to Missouri and the West. Ranking MG in the Army when named by Lincoln 14 May, '61. this was not necessarily unfounded as he had previous military experience and was a major factor in securing California to the United States in the Mexican War. Commanded in Missouri in '61 and West Virginia in '62. Once removed, political differences with Lincoln led to his not receiving any further commands;
Andrew Johnson- Tennessee senator who remained loyal and Lincoln's 2nd term VP, shortly afterwards succeeding to the presidency upon Lincoln's assassination. Named BG and military governor of Tennessee March '62 to March '65 charged with succorring the Unionist element of Tennessee and running roughshod over partisans in the state, which he left to commanders in the field;
James Henry Lane- CW senator from Kansas, MG and leader of the Kansas militia and the notorious Jayhawkers, an irregular who fought irregulars in Kansas and Missouri in the irregular fashion noteworthy in the Trans-Miss. Helped blunt Price's Missouri Raid of '64;
John Alexander McClernand- prominent Illinois War Democrat, US Congressman of long standing, and another close friend of Lincoln's, MG, commanded divisions,corps, and the self-styled Army of the Mississippi during the Vicksburg Campaign. Fought in the major battles in the cis- Mississippi West till removed by Grant after the assault of the Vicksburg fortifications. A militant anti- West Pointer, was a major thorn in Grant's side in his western campaigns;
Robert Huston Milroy- prominent Indiana militiaman and judge, a MG, held division and district command mostly in Ky/Ten and western Virginia, known for his depradations in the Shenandoah and western Virginia, crushed by Ewell at Winchester at the start of Lee's Gettysburg Campaign when he rashly thought he could hold the town against Lee's legions;
Daniel Edgar Sickles- a Tammany Hall politico, became a fierce War Democrat and later Radical Republican, MG, raised a brigade at the war's commencement at his own expense and held brigade, division, and corps assignments in the Army of the Potomac until losing his leg in the Peach Orchard at Gettysburg. Denied command upon his convalescense by Meade and Halleck, he became a troubleshooter for Stanton till the war's end, then becoming a postwar military governor in the defeated South. Unliked by the professional soldiers of the Army of the Potomac, but a favorite of Stanton. Had a long running feud with Meade particularly over Gettysburg. Later as a US Congressman, he was instrumental in creating the Gettysburg National Military Park.
Please note I left out any leader out of West Point or any prominent political men who had seen extended service in the Army or commanded at high rank in the Mexican War (John A Dix, Robert Patterson) or had been educated in academies, held ohigh office, or been active in foreign wars (Franz Sigel, Carl Schurz.) A number of the men listed had previous military experience- Milroy had been a captain and Lane, Baker, and Fremont had held the rank of colonel in the Mexican War. I list these men here as it is primarily their political influence which led to their important commands in the Civil War- Fremont most notably, who became if I am not mistaken the senior ranking officer in the Army with his appointment, which was to the US establishment and not to the volunteers and as he was the most prominent politician of military credentials in the Republican Party.
I am also leaving out generals who would qualify for this list who were some of Lincoln's successes. They held command at the corps level, but not higher ( Francis P Blair, John A Logan, Robert C Schenck). These men mostly commanded in the loose and informal West, where under the easy going tutelage of the likes of Grant, Halleck, Sherman, Thomas, and McPherson, they found success that seemed to evade their Eastern based brethren. As it was, professional men who had not fluorished in the East managed to thrive in the West as well, Oliver O Howard and Henry W Slocum come to mind.