Most of the volunteers who joined up at the early part of the war did so on three year terms. And although many chose to re-enlist and finish the fight, plenty more decided they had done enough and returned home in 1864.
How were these men treated by their communities? Was there a sense that they had served with honor and should be welcomed back with as having done their part and now it was rightfully someone else's turn, or was there a sense that they were running away from a job not yet finished, and subject to community censure?
How much would these reactions vary geographically? Would a New England community view a man differently than one in the Midwest under the same circumstances? What about politically? Were Democratic strongholds more understanding of the non-reenlistments versus areas which were heavily Republican?
How did other veterans themselves view such men? Especially during the war as they stayed around and saw these men leaving, but also in later years at reunions and in general commemoration of the war?
I've done a lot of research into Civil War veterans and I haven't found any instances of volunteers facing stigma for not reenlisting while the war was going on. There was, however, tremendous pressure to reenlist if you were one of the short-term volunteers during the beginning of the war (30-day, 60-day, and 1-year volunteers), but the impression that this had more to do with the brevity of these contracts.
The fact that the Army had some difficulty retaining veterans also suggests soldiers faced few negative social effects for not reenlisting. Soldiers were often induced to reenlist with the promise of promotions and bonuses (it's the same in today's Army), and no one held it against you if you opted to leave the military. Many other soldiers reenlisted because they were able to go home and spend a few months on recruiting duty—some even spent their entire second enlistments in cush jobs.
One also has to consider the culture of volunteer militarism during the Civil War era. The regular Army was small because most people thought a large standing Army was (1) too expensive to maintain, (2) unnecessary due to the country's relative isolation, and (3) a threat to liberty and democratic republicanism. Conversely, men felt obligated to volunteer in the Army in times of war, and American volunteerism always took the terms of service contract/obligations very seriously (British officers during the French and Indian War in the 1750s used to accuse American volunteer militiamen of being unpatriotic because they insisted on serving only how and as long as described in their "contracts"). As a result, American society tended to treat deserters who violated their terms of service badly (even this has some caveats because many communities were divided in how to treat deserters), volunteers who completed their enlistments were seen as doing their part.
Also consider this—if a volunteer signed up to serve in the U.S. Army for three years in late 1861 and early 1862 (when most three-year contracts began), they were due to leave the military in late 1863 and early 1864. If you served in the field (as opposed to garrisoning a fort away from combat), you had experienced some hard service. On top of that, the draft started in 1863 and the Union Army was only getting larger. While there was a need for experienced non-commissioned and commissioned officers, soldiers leaving the Army after serving out their enlistments wasn't an existential threat to the United States. That was totally different in the Confederacy, where the army was incurring huge and irreplaceable casualties and desertion, and the government had to pour a tremendous amount of resources in keeping men into the military. In fact, by mid-1864 the Confederate army was comprised largely of draftees and die-hard true believers, while the U.S. army was augmenting a steady stream of fresh white volunteers with draftees and black volunteers.
In short, regardless of where you lived in the North, if you completed your initial enlistment honorably (especially if it was a 3-year stint), most pro-Union people thought you had done your part.
P.S.—As a historian of the Civil War in the Far West, I would be remiss if I didn't mention reenlistments among Californian and Nevadan soldiers. Most of these men signed three-year terms in 1862, and it was clear that the vast majority weren't going to reenlist as early as 1863. Most hated the idea of re-upping just to fight Indians and garrison forts in the frontier, so most of these volunteers were planning on returning to California or Nevada, or traveling east to serve in the "seat of the war." To prevent a mass exodus from the volunteer regiments, the Army offered these men bounties of anywhere from $50 to $150 to reenlist. This was a lot of money, considering a private in the Union army made $13 a month (they got a $3 raise in 1864—huzzah!). They also got to be part of special "Veteran" regiments and got to wear cool red lines on their sleeves to show the noobies they were special. All they had to do was sign an enlistment extension. Many of these volunteers continued to serve well into 1866, when they were finally replaced with regular Army soldiers after the Civil War ended.
I can say for a fact that these soldiers faced zero negative social ramifications if they didn't reenlist. Hell, it's questionable if they were even stigmatized for deserting the Army.