The new cat starts behind a closed door
Set up one room for the newcomer with her own litter tray, food and water bowls, bed, and scratching surface, and bring her home straight into it. Your resident cat keeps the rest of his territory, his routine, and his access to you unchanged. That separation is not a delay to the introduction; it is the introduction, because the first thing each cat learns about the other should be a smell drifting under a door, investigated at leisure, with no face attached to defend against.
Give the resident cat extra of your time during this stage, in his usual places. A cat who feels displaced by the smell behind the spare-room door, and who also loses his evening lap time, has two grievances, and he will attach both to the newcomer.
Scent before sight, always in that order
After a day or two, swap bedding between the rooms so each cat sleeps with the other's scent, and stroke each cat's cheeks with a soft cloth to place by the other's bowl. Feed both cats on their own side of the closed door, bowls set back far enough that eating feels safe, then gradually closer over several days as both eat calmly. Eating well near the door, then sniffing under it and walking away, are the green lights for the next stage. Hissing at the door in the first days is normal information-gathering; a cat who still cannot eat anywhere near it after a week is telling you to slow down, not to push on.
First sight through a barrier, briefly
Move to visual contact with a physical barrier between them: a slightly opened door secured so neither can slip through, or a tall pet gate with a sheet over part of it so each cat can choose how much of the other to look at. Keep these sessions short, feed or play with each cat in sight of the other, and end while things are still calm. What you want to see is indifference developing: a glance, then back to the toy. Fixed staring, a low growl, ears flattening, or a lunge at the barrier means the session ends and you drop back a stage for a few days without treating it as a failure.
Shared space, with escape routes and duplicates
The first shared sessions should happen in the largest, least contested room, with doors open so either cat can leave, high perches available, and you playing with each cat at opposite ends. Scatter resources so nothing needs contesting: an extra litter tray beyond the rule of one per cat plus a spare, separate feeding spots, and more than one good resting place.
Brief scuffles with a hiss and a retreat are different from a real fight with screaming, fur flying, and one cat cornered. Break up a genuine fight with a loud noise or a cushion placed between them, never with your hands, then separate the cats and return to the barrier stage. If aggression is intense, repeated, or one cat stops eating, toileting normally, or starts hiding all day, bring your veterinarian into the conversation for health and behaviour guidance.
Steady stages need steady hands
Introductions fail at weekends, when a keen family member skips three stages because the cats seemed fine at breakfast. Agree the stages, who runs each supervised session, and who records how both cats ate and behaved, so progress is judged on evidence. KinsLe's Care Teams lets you assign those feeding, scent-swap, and supervision tasks for each cat to specific people, with each completed session marked off, keeping the introduction at the cats' pace rather than the household's enthusiasm.