When modifying bullets in a given load, you must pay attention not only to bullet weight and construction; but also to bullet shape, in particular the style of ogive.
This can have a significant effect on the distance of contact of the bullet to the lands. And this in turn, of course, can have a dramatic effect on pressures generated!
In your query, you mention using Hornady bullets instead of Speers. Hornady's typically have secant ogives, which places contact distance to the lands further back than the tangent ogives that Speer bullets typically exhibit.
(In this case, your pressures should be lower) But if you are going from a secant ogive to a tangent ogive, you could experience an unwelcome pressure spike! Depending upon the bullet seating depth and the chamber of your rifle, you could even be jamming the bullet into the lands.
So, the advice to start off low, and work your way up carefully definitely holds, but I would add that you also check your bullet's distance off the lands.