First: is it blocked?
Before anything else, the flue has to be confirmed clear. Collapsed tile, nesting material and heavy creosote all restrict flow enough to push smoke back into the room, and this is both the most common cause and the most serious.
This is a safety check, not a comfort check. A blocked flue serving any fuel-burning appliance needs clearing before the appliance is used again.
Second: opening-to-flue ratio
A masonry fireplace needs roughly one part flue area to ten parts opening area. Plenty of fireplaces were built with openings too large for their flue and have smoked since the day they were finished.
Where the flue cannot change, reducing the effective opening height with a smoke guard restores the ratio and often fixes a lifelong problem.
Third: the smoke chamber
Above the damper, older chimneys were built by corbelling brick inward, leaving a stepped and mortar-smeared surface that slows and cools gases.
Parging that chamber smooth with refractory mortar is one of the most underrated draft improvements available, and it reduces creosote accumulation at the same time.
Fourth: the house is winning
A modern airtight house with a powerful range hood, bath fans and a dryer can be under enough negative pressure that the chimney becomes the easiest air inlet. Air comes down the flue and takes the smoke with it.
Test it: open a window near the fireplace and see if the smoking stops. If it does, the answer is a makeup air strategy rather than chimney work.
Fifth: stack effect and cold flues
An exterior chimney full of cold air resists the first fire of the day. Priming the flue with a rolled newspaper torch held up at the damper for a minute reverses the flow before you light the fire proper.
A top-sealing damper helps here too, because it keeps the flue warmer and drier between fires.