A few things separate the creative leads who make this work from the ones who churn through three partners in a year.
Route by type of work, not by overflow. The failure mode is sending the partner whatever's on fire this week. That just recreates the freelancer problem. Instead, draw a clean line: internal team owns product design and brand direction, the partner owns marketing and conversion surfaces, or whatever split fits your context. Clean lanes mean less briefing and better work in both lanes.
Insist on senior designers, not a production pool. If the external work needs your correction every round, you've added a management burden, not capacity. The whole point is work you can trust without babysitting it. Ask directly who does the work and how long they've worked with SaaS companies specifically.
Keep the commitment flexible. Your capacity problem is real today, but it will change shape after your next hire. A subscription model with no long-term contract lets you scale the extension down when the team grows, instead of paying for a retainer you no longer need.
And position it internally as your decision, made to protect the team. Said plainly to leadership: "I'm adding senior capacity now so we stop losing conversion work to the backlog, and so we don't burn out the team before the next hire lands." That's a creative lead running resourcing strategically. It reads as strength, not as a gap.