performance tuning - Given an ordered set S, find the points in S corresponding to another list

Given a sorted list of numbers $S$, I want to create a function that accepts a list of numbers $L$ and for each number $l \in L$ it returns the index of the largest number $s \in S$ such that $s

SeedRandom[13];
S = Sort @ RandomReal[10, 5]

{0.405196, 4.56535, 7.04274, 7.95001, 8.6823}

And, here are a couple examples of the argument to the function:

SeedRandom[10];
list1 = RandomReal[10, 5]
list2 = RandomReal[10, 3]

{6.67917, 8.33874, 4.61316, 4.83263, 9.52033}

{6.0669, 1.22425, 6.13959}

Then, I want to create a function f:

f = findIndices[S];

such that

f[list1]
f[list2]

return:

(*

{2, 4, 2, 2, 5}
{2, 1, 2}
*)

One possibility is to use:

findIndices[s_] := Interpolation[
    Thread[{s, Range@Length@s-1}],
    InterpolationOrder->0,
    "ExtrapolationHandler" -> {Evaluate[Length[s]]&, "WarningMessage" -> False}
]

But, this approach is quite slow when dealing with large arguments:

f = findIndices[S];

tst = RandomReal[{.5, 10}, 10^6];
f[tst]; //AbsoluteTiming

{1.15025, Null}

I'm interested in arguments on the order of 10^6 elements, and ordered sets $S$ on the order 10^4 elements. Is there a faster method?