You'll have to be more specific about the reactions involved here; some obvious parameters which affect conversion rates would be
a) Reduce the concentration of the product species that may be in the feed - that will drive the reaction forward - there may be some disadvantages to doing this in some cases
b) Increasing reaction temp generally increases conversion, regardless of whether the reaction is endothermic or exothermic; but selectivity for the desired product will always decrease.
c) If the reaction is pressure dependant, then changing the reactor pressure in the right direction can increase conversion
d) Reduce duration from SOR to EOR to get a higher average catalyst activity
e) Change to a more stable catalyst with higher activity, with less sensitivity to poisons in the feed
f) Decrease space velocity
Others will add on more no doubt