This is a hand from a little while ago, I think villain was unknown. In the moment I decided to make the call souly based on the fact that I had the queen of spades blocking his flush combos. Not sure if queens is in the call range here or not.

Hand #62329049 - Holdem(No Limit) - $0.02/$0.05 - 2019/06/15 04:50:06 UTC
South Holland 6-max Seat #3 is the button
Seat 1: saking ($4.81)
Seat 2: EasyMoney420 ($5.08)
Seat 3: Greenerrr ($5.64)
Seat 4: iAitchoP ($5.31)
Seat 5: tmorellopoker ($5.07)
Seat 6: mexicanmango ($7.42)
iAitchoP posts the small blind $0.02
tmorellopoker posts the big blind $0.05
*** HOLE CARDS ***
Dealt to Greenerrr [Qs Qc]
mexicanmango calls $0.05
saking folds
EasyMoney420 folds
Greenerrr raises $0.20 to $0.20
iAitchoP folds
tmorellopoker folds
mexicanmango calls $0.15
*** FLOP *** [Kd 9s 6s]
Main pot $0.45 | Rake $0.02
mexicanmango checks
Greenerrr bets $0.40
mexicanmango calls $0.40
*** TURN *** [Kd 9s 6s] [2h]
Main pot $1.21 | Rake $0.06
mexicanmango checks
Greenerrr checks
*** RIVER *** [Kd 9s 6s 2h] [Ks]
Main pot $1.21 | Rake $0.06
mexicanmango bets $1.27
Greenerrr calls $1.27
*** SHOW DOWN ***
Main pot $3.62 | Rake $0.19
mexicanmango shows [Qd Kh] (three of a kind, Set of Kings [Ks Kh Kd Qd 9s])

i wouldnt worry about "what your range" is supposed to do at these stakes, bkuz for that to work villain has to be putting you on a range.
i wouldnt bet the flop that large with an under pair, id bet 33-50% pot then slow down if called, like u did.
river im calling stabs but an overbet is like.. pretty much always the nuts at these stakes brot its an easy fold .