Looks like I'm being a little loose with the original topic, but the following examples show the time lines involved in communications regarding resignations.

Note one example of a communication from Little Rock to Richmond is dated Nov. 25, and then marked "received" in Richmond December 6. That's considerably faster than I would have guessed.

Its interesting to note how Holmes was worried about the "great number of resignations" coming from Hindman's new 1st Corps.

The following correspondence and orders are taken from the Official Records and paraphrased extracts from the Alexander Papers.

August 16, 1862, Special Orders Army of the South West #68
Lt Colonel Ringo's resignation is accepted
Major C H Rumdell's [cmdg. Arty battalion] resignation accepted

Oct 22, 1862-Hindman to Rains
Your resignation is accepted from this date

Oct 23, 1862 endorsement
Camp near White River
approving Col S Jackman tendering resignation and authorizing him to raise a cav regiment in Missouri

Oct 25, 1862
Special Order #22 Trans Miss District
Camp War Eagle
.............................
Colonel Stevens in arrest
Colonel Coffee under arrest
Colonel Jackman of Jackman's regiment of Missouri Infantry..... resignation accepted.
....................

Nov 4, 1862
from Hawpe's regiment of Texas cavalry
resignation of Col Hawpe for disability

Nov 6, 1862
from McRae's brigade camp on Mulberry
resignation of Col McNeill

Nov 6, 1862
from Little Rock....copies Special Order #77, paragraph seven
accepting Lt Colonel Ringo's and Col Johnson's resignations

Nov 7, 1862
from McRae's brigade camp
resignation of Colonel Matlock

Nov 8, 1862
Shoup's division
Major Crenshaw's resignation of Grinstead regiment

Nov 12, 1862
Headquarters, Trans Mississippi Department . Little Rock
accepting resignations of .... McNeill, Matlock, Major Crenshaw Capt Ballard company E, Caldwell's regiment, Colonel Hawpe

OR Vol 13 HEADQUARTERS TRANS-MISSISSIPPI DEPARTMENT,
Little Rock, Ark., November 13, 1862.
Major General T. C. HINDMAN,
I am instructed by the major-general commanding to call your attention to the great number of resignations being sent in from your command, and to request you will scrutinize closely all such papers as may come before you, and not recommend such to be accepted unless perfectly satisfied in your own mind that they are based on certain and sure benefit to the service on purely military grounds. With regard to the resignation of the higher officers, he wishes you to look to it particularly that they do not offer their resignations from some personal motive, fancied injury, dissatisfaction, pique, or some other like cause that might be harmonized, for it is better such differences should be harmonized than the parties turned loose, leaving a party behind them without the different being healed, and consequently the object sought to be gained not accomplished.
......................

Nov 15, 1862
Hindman to Holmes regarding the number of resignations
.............some are due to being poor officers according to the examination board, some honestly admitted not being up to it, some are due to physical problems. Exceptions are General Rains, Colonel Johnson, Lt Colonel Ringo.
Rains was given his rank on condition he would resign if drunk...........[ and he got drunk]
Johnson is awful and was never with his command-originally his was the best of the new Texas regiments, now the worst he resigned at my request.........
no one liked Ringo... men or officers and he was thus a loner and had no idea of discipline... He also resigned at my request

Nov 24, 1862 hdqrs. Shaver regiment
letter from Lt Colonel W C Adams inquiring as to his resignation which has been forwarded

OR Vol 13
LITTLE ROCK, ARK., November 25, 1862.
(Received December 9, 1862.)
General S. COOPER, Adjutant and Inspector General C. S. Army:
................................................... General Hindman's infantry force is probably equal to this, and is now near Fort Smith, where it can be subsisted and the demoralized troops reorganized and reassured under officers who will restore their confidence. ......................As I wrote to you, General Rains, for his drunkenness when the enemy advanced from Missouri, has been directed to resign. I hope his resignation may be accepted, as the shortest way of getting rid of him. If the witnesses can be had General Cooper will be brought before a court of inquiry under Orders, Numbers 38, through he denies the charges made against him by Captain Mackey, C. S. Engineers, emphatically. Mr. Boudinot writes to me that our Indian relations will soon be in a satisfactory condition and the Indians satisfied..................... On the arrival of Major-General McCown I will leave him here and pay the Indians a visit. . ............................
General Holmes