We have a variation which we call "Nash Whist". (Disclosure: Our group of players has been refining "Solo Whist" for many years. I have documented our version at the linked location.)
We have streamlined and simplified the shuffling and dealing (as described in the earlier answer).
In the original game the bids are limited. Apart from the Misere bids, the solo bids are Solo (five tricks) and Abundance (nine tricks) etc. This means that the main bid that occurs is Solo, and you also get paid for any "over-tricks" that you manage to make beyond the five.
For Nash Whist, we added other bids at every level from 6 to 13. We also added rank of suits, as for Bridge (explained below) and No-trumps.
Apart from the normal Prop-and-Cop, the minimal bid is the normal "Five-in" solo, i.e. play alone against the other three with the initial turned-up suit as trumps. So win five tricks in the indicated trump suit.
The trump suit can be changed by any bidder when it is their turn to bid.
Higher bids will beat the previous bid. For example "Seven-Out" in any suit will beat "Six-In" (i.e. Six in trumps) and Seven-In will beat Seven-Out.
The rank of suits is as for Bridge (i.e. Spades is highest, then Hearts, Diamonds, Clubs).
For example, if Hearts was turned up trumps and the current bid is Six-In, then a bid of Seven-Out Clubs would beat that, and a bid of Seven-Out Diamonds would beat that. Note that a bid of Six-Out Spades does not beat Six-In, because in this example Hearts are trumps and so automatically rank at the top.
The bidding can start at any level, and proceeds with higher bids until no-one wants to bid further. The last bid is the "declarer". Their suit is now Trumps.
There are no payments for over-tricks. This encourages people to bid to the limit of their hand and not be safe.