Try infernal calling, although you'll need the imp's talisman
The spell infernal calling (XGtE, p. 158) also allows the summoning of a particular devil. The spell usually just allows you to specify the CR, meaning that there's no guarantee that you'll get a specific devil:
You choose the devil's type, which must be on of challenge rating 6 or lower [...]
At Higher Levels. When you cast this spell using a spell slot of 6th level or higher, the challenge rating increases by 1 for each slot about 5th.
So you can summon a devil of up to CR 10 (if cast with a 9th level spell slot). However, you specified an imp, so that's well within the CR range.
However, later in the spell's description, it goes on to say:
If you possess an individual devil's talisman, you can summon that devil if it is of the appropriate challenge rating plus 1, and it obeys all your commands, with no Charisma checks required.
So, if you can somehow get hold of this particular imp's talisman, you can summon that specific imp with infernal calling. This is still a 5th-level spell, so not that much lower than your stated options, but it's at least a Tier 2 level spell rather than a Tier 3/4 level spell, and is probably the most reliable method without needing access to 9th-level spells (although it does require the imp's talisman).
There isn't much information out there about these talismans, since I imagine it's intentionally left vague so that DMs can fill in the blanks themselves and decide how such things work in their worlds. However, in the Monster Manual, there is a little additional information on these talismans (p. 67):
Devil True Names and Talismans
A mortal who learns a devil's true name can use powerful summoning magic to call the devil from the Nine Hells and bind it into service. Binding can also be accomplished with the help of a devil's talisman. Each of these ancient relics is inscribed with the true name of a devil it controls, and was bathed in the blood of a worthy sacrifice - typically someone the creator loved - when crafted.
Only imps are truly content to be summoned and they easily commit to serving a summoner as a familiar, but they still do their utmost to corrupt those who summon them.