An outsourced role has a structural ceiling and there is no honest way around it: the person's employer is the provider, the interesting work belongs to the client, and promotion in the usual sense means leaving the account.
Saying it plainly
Clients sometimes promise progression they cannot deliver. You cannot promote somebody who does not work for you, you cannot give them a title in your organisation, and you should not imply otherwise.
What you can do is real and it is worth more than most people assume.
Widening rather than raising
Give the person a whole process rather than a set of tasks. Ownership of something end to end, including the decisions in it, is the closest available thing to promotion and it is entirely within your gift.
Let them improve the documentation and treat their changes as the authoritative version. Let them define how the work is done rather than only doing it.
Training the next person
When the team grows, the person who has been doing the work trains the newcomer. It is better training than yours, it recognises their expertise publicly, and it changes how they describe their own job.
It also produces the documentation test from the entry on writing things down, at no cost.
The skills question
Ask what they want to learn. A person who wants to move from bookkeeping to analysis, or from support to quality assurance, can frequently be given a slice of it inside their existing role.
The alternative is that they learn it somewhere else, which they will, and then work somewhere else.
What the provider can do
Team leader tracks, cross-account moves, formal training, and paths into supervision or operations. These are theirs rather than yours and they are worth asking about before signing.
The conversation to have annually
Where do you want to be in two years, and what would get you there. It is an ordinary management conversation and outsourced staff are rarely asked it by anybody.
You will sometimes hear that they want to do something your account cannot offer. That is useful information and it is better received eighteen months before a resignation than three months after one.
Titles
A title inside the account costs nothing and is not nothing. Senior, lead, specialist: these appear on a curriculum vitae, they are used inside the office, and they mark that somebody has progressed even though they sit in the same seat.
Agree it with the provider, because the title has to be consistent with their own structure, and then use it.
Pay, briefly
Progression without any movement in pay is noticed. The entry on pay expectations covers what a client can and cannot do about it, and the short version is that you should establish before you need to know whether you can fund an increase.
The honest ceiling
Somebody genuinely excellent will eventually outgrow an outsourced role, and the good outcome is that they move on well: to a team leader role with the provider, or to something else entirely, with a reference and a handover.
Trying to hold them produces a resentful year and then the same departure. Helping is cheap, it is remembered, and the industry here is small enough that being known as a good client to work for has consequences in recruitment.
The one that costs nothing
Ask their opinion about the work and act on it visibly. Somebody whose suggestion changed how a process runs has been promoted in the only sense available to them.