The two lower-league clubs are vying for a second-round placement with Tottenham Hotspur Under-21s and Bristol Rovers, who have already played out a thrilling 3-3 draw in the first phase.
Match preview
Despite a handful of semi-final appearances, Exeter City are still searching for their inaugural EFL Trophy honour and have several wrongs to right from previous tournaments this week, having suffered group-stage eliminations in each of the past two seasons.
Now in their third straight season of League One football under Gary Caldwell, the Grecians' opening to 2024-25 has been the true definition of a mixed bag, having marched to a pair of victories over Rotherham United and Bolton Wanderers either side of consecutive losses to Northampton Town and Peterborough United.
Saturday's worthwhile trip to Bolton saw Kamari Doyle and Millenic Alli strike either side of the half-time whistle for Caldwell's troops, who avenged last season's horrendous 7-0 defeat at the hands of the Trotters, leading the boss to hail his side's 'superb' defending from the front.
Having lifted that curse at the Toughsheet Community Stadium, Exeter now endeavour to exorcise some more recent demons, having lost to Walsall on penalties in their opening EFL Cup tie of the season; a 12-yard shootout will take place on Tuesday if the tie ends level.
EFL Trophy success has also been extremely difficult for Swindon Town to come by in the 21st century - with the possible exception of losing the 2011-12 final to Chesterfield FC - and they are also aiming to avoid a third successive group-stage exit this time around.
In fact, only one of the visitors' last seven appearances at the competition has seen them advance to the second round, and the exploits of Mark Kennedy's side so far in 2024-25 suggests that a deep EFL Trophy run may be a thing of fantasy.
Indeed, Swindon are still on the hunt for their first win of the campaign after five matches, having drawn two and lost two of their opening four League Two matches in addition to a 4-2 defeat to Crawley Town in their opening EFL Cup affair.
Kennedy's men were in the ascendancy against Notts County on Saturday thanks to a Gavin Kilkenny strike, only for David McGoldrick to turn the tie on its head with a heroic brace, but the Swindon boss was still able to reflect on the pleasing aspects of his side's display.
However, there have been few positives for Swindon to take from their previous clashes with Exeter, who have emerged victorious from the last four matches between the two clubs, including a 1-0 win in the group phase of last season's EFL Trophy owing to a Harrison Minturn own goal.
Exeter City form (all competitions):
Swindon Town form (all competitions):
Team News
A late Exeter arrival in the summer window, Leicester City-owned winger Amani Richards is ineligible for Tuesday's game, having played - and scored - for the Foxes' Under-21s earlier in this year's EFL Trophy.
Richards was only a second-half substitute in the win over Bolton anyway, and Caldwell would presumably hope to stick with that same winning formula for Tuesday, unless fresh injuries or jellied legs force his hand.
Experienced winger Yanic Wildschut will spend months on the sidelines with an ACL injury, though, while Josh Magennis could not complete the full 90 at the weekend; Sonny Cox is waiting in the wings.
In contrast, none of Swindon's new recruits are cup-tied for their first EFL Trophy contest of the new term, while Ryan Delaney and Sean McGurk are back in training as they step up their recoveries from respective injuries.
Kennedy's side emerged unscathed from their narrow loss to Notts County, though, and even though the visiting coach was proud of his side's battling display, changes should be afoot.
Kabongo Tshimanga and Daniel Butterworth can both provide an alternative presence in attack to former Liverpool striker Paul Glatzel, who remains without a goal five games into the new season.
Exeter City possible starting lineup:
MacDonald; Fitzwater, Crama, Sweeney; McMillan, Doyle, Cole, Purrington; Aitchison; Alli, Cox
Swindon Town possible starting lineup:
Barden; Hall, Minturn, Wright; Sobowale, Clarke, Cotterill, Kilkenny, King; Butterworth, Glatzel
We say: Exeter City 1-0 Swindon Town
Keeping things tight in defence remains a major problem for Kennedy's Swindon, who ought to be fighting an uphill battle against an Exeter side revelling in a revenge win over Bolton.