In other words, would an increased tension increase tuning stability with the use of the tremolo?

I understand that this would stiffen the feel of the tremolo, but providing that all other parameters remain as per recommended measurements (plate at 1/8'' off the body, claw screws to suit, etc.) would the increased tension help?

If you bend, it will help keep the other strings in tune as the bend changes the total tension on the remaining strings.

That's precisely one of the reasons why I'm thinking of putting another spring.

But I was also wondering if the guitar would hold tuning better when the tremolo is used...

Yes. However, it is but one factor in a chain of other factors such as but not limited to, the correctness of the nut cuts, the straightness of the neck, the overall action, the quality and correctness of the winding of the strings, bridge saddle quality and height, pitch of the tremolo block, height of the pickups, graphite lubrications as and where required, the guitar's environment, personal technique, etc, etc.

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You dig?

Yes it will hold better tune but you will limit the amount of tremelo and control you have