Like most maladies, it's best to avoid the risk factors rather than just treating the symptoms. Not to say that there haven't been thoughtful responses posted on relieving the pain.
Cumulative trauma, such as tennis elbow is brought about by all the usual suspects: high repetition (casting)' high force ( especially grip), awkward postures ( primarily the wrist). All the risk factors are combined when getting the fly out there.
It's alot less of an issue while summer fishing for trout, bass, and lighter fish that don't require heavy rods, big sink tips or heavy flies. But salmon and winter steelhead fishing require greater strength and alot more casts.
The extensor carpi radialis brevis is generally the muscle/ tendon that is involved. It attaches on the lateral(outside) of your elbow. With a really symptomatic tennis elbow, it hurts to even touch that spot.
That muscle among others helps to extend your wrist. If your hand is flat side down on a table, it will assist to bring it up against gravity, as an example. So in casting its very involved in bringing the rod back when first drawing the fly off the water before starting your next cast, and just before starting the forward casting stroke.
The forearm rotators attach on the medial and lateral epicondyle and can be pain generators as welland are are involved in the cast.
But the hidden factor is how the extensive carpi radialis brevis is involved in grip. I think grip is likely the biggest risk factor. The extrinsic
finger flexor muscles, the ones that bend your fingers, attach at your elbow but on the medial (inside) of your elbow. So why does the outside of your elbow hurt with forceful grip?
The extensor carpi radialis brevis has to fire every time we make a fist. If it didnt happen automatically, our wrist would flex ( bend towards the direction of your palm) with every making of a fist. The finger flexors cross the palmer side of wrist and so also act as wrist flexors. So the stronger the grip an activity requires, the greater force with which the exterior carpi radialis must fire to keep the wrist from bending in. Try gripping something heavy with your wrist flexed inward. You'll notice the loss of strength.
So again,no big deal with lighter fly rods, floating leaders, light flies, less casting, etc. Decreased risk factors there.
So what to do? If salmon/ steelhead fishing with heavier rods, maybe try using two hands at least in drawing the fly out of the water before starting your next cast. The water tension combined with a heavy fly/ sink tip is culptit number one in my opinion. And maybe consider learning to spey cast with two handed rods this summer at your leisure, before the next salmon steelhead season. Two hands cuts the force considerably, evn with the bigger spey rods.
Take more breaks.
Use the lightest rod, sink tip, fly size you can get away with for steelhead/ salmon for your single hand casting.
Prior to the start of next season, condition your grip, wrist and forearm muscles. Ask your doctor, therapist or look on line for exercises. These exercises should be comfortable, and build both strength and endurance. If your activities require a lower percentage of your available strength, it lowets the risk of cumulative trauma. So strong is good, but don't over do it.
If your tennis elbow is active, best to avoid fishing for now. But if you must, you must. But dont overdo the cures.
Disclaimer-- I'm just trying to provide information on some causes and risk factors for the condition. Always check with your treating health care provider before proceeding with anything in my post. They will know your full medical history and any reasons why my suggestions are perhaps not relevant or advisable.