Q:

Increase the range of Dissonant Whispers

I have a level 10 bard, who is looking to make some assassinations during the course of the campaign.  Dissonant whispers has the nice property of only the target hearing the whisper itself. 
However, it has a range of paltry 60 feet, which is uncomfortably close in many situations, especially if DM makes a harsh ruling about the audibility of the verbal components of the spell (even if the actual whisper is inaudible). 
Is there a way to increase the range of Dissonant Whispers? 
Multiclassing Bard/X is possible but not preferable.  Other options I can think of are magic items and feats. 
Preferred sources are the core books and Xanathar's Guide to Everything.  Setting is Exandria, so at least the Tal Dorei and Wildemount books would be acceptable too. 
If the method applies to other verbal-only saving throw spells of the Bard, all the better of course. 
Note: The Spell Sniper feat does not apply, as it requires a spell with an attack roll. 

A:

Multiclass Sorcerer
So far as I know there are no magic items that do what you want.  Spell sniper does not apply as you found yourself. 
Multiclassing sorcerer seems the only option from sourcebooks.  At sorcerer level 3, you get the metamagic feature for which you can choose the distant spell which will double your range. 
Multiclassing prerequisites should not be a problem, as a bard you will have high charisma anyway. 
As Someone_Evil suggested in the comments, you can gain the distant spell option with the Metamagic Adept feat from the Unearthed Arcana: Feats from July 2020.  This depends on UA being allowed in the game, however.  Also, this gives only 2 sorcery points. 
Obviously, these options depend also on how often dissonant whispers needs to be cast.  If you have to reduce the RAW hit points of a victim, the sorcery points won't suffice.