How can I label segments of a smooth curve through some nodes?

I have some nodes (some of which are calculated using the intersections library) and want to draw a smooth curve through some of those nodes, labeling the curve between each node pair.

The draw plot[smooth] coordinates {...} command draws a nice smooth curve.
How can I label the segments of that curve?

The only workaround I could come up with is drawing the straight lines in white, so that only the labels are visible, but I’m sure there is a better solution, since that way label positioning relative to the smooth curve varies greatly (e.g. label 3 is far from the curve whereas label 4 is very close).

documentclass{article}
usepackage[utf8]{inputenc}
usepackage{amsmath}
usepackage{tikz}
usetikzlibrary{calc}
usetikzlibrary{intersections}
usetikzlibrary{positioning}

begin{document}
tikzstyle{vertex}=[circle,draw=black!30,fill=black!30,inner sep=1pt]
begin{tikzpicture}[scale=2]

node[vertex] (a) at (0,0) {};
node[vertex] (b) at (2,-.3) {};
node[vertex] (c) at (2.5, 1.3) {};
node[vertex] (d) at (3.5, .6) {};
node[vertex] (e) at (3, .2) {};

draw (a) -- node[above,pos=.2] {1} (b) -- node[left] {2} (c) -- node[below left] {3} (d) -- node[below] {4} (e);
draw[red] plot[smooth] coordinates {(a) (b) (c) (d) (e)};

end{tikzpicture}
end{document}

documentclass{article}
usepackage[utf8]{inputenc}
usepackage{tikz}
usetikzlibrary{calc}
usetikzlibrary{decorations.pathreplacing}
newcounter{smoothcounter}
begin{document}
begin{tikzpicture}[scale=2,
    vertex/.style={circle,draw=black!30,fill=black!30,inner sep=1pt},
    label smooth/.style={decorate,decoration={show path construction,
        curveto code={
path (tikzinputsegmentfirst) .. controls
(tikzinputsegmentsupporta) and (tikzinputsegmentsupportb) ..(tikzinputsegmentlast) 
coordinate[pos=0.4] (aux1) coordinate[pos=0.6] (aux2)
(aux1) to[edge label={$stepcounter{smoothcounter}arabic{smoothcounter}$}] (aux2);
}}
}]
  node[vertex] (a) at (0,0) {};
  node[vertex] (b) at (2,-.3) {};
  node[vertex] (c) at (2.5, 1.3) {};
  node[vertex] (d) at (3.5, .6) {};
  node[vertex] (e) at (3, .2) {};
  draw (a) -- node[above,pos=.2] {1} (b) -- node[left] {2} (c) -- node[below left] {3} (d) -- node[below] {4} (e);
  setcounter{smoothcounter}{0}
  draw[red,postaction=label smooth] plot[smooth] coordinates {(a) (b) (c) (d) (e)};
end{tikzpicture}
end{document}

documentclass{article}
%url{https://tex.stackexchange.com/q/570505/86}
usepackage[utf8]{inputenc}
usepackage{amsmath}
usepackage{tikz}
usetikzlibrary{calc}
usetikzlibrary{intersections}
usetikzlibrary{positioning}
usetikzlibrary{hobby}

begin{document}
tikzstyle{vertex}=[circle,draw=black!30,fill=black!30,inner sep=1pt]
begin{tikzpicture}[scale=2]

node[vertex] (a) at (0,0) {};
node[vertex] (b) at (2,-.3) {};
node[vertex] (c) at (2.5, 1.3) {};
node[vertex] (d) at (3.5, .6) {};
node[vertex] (e) at (3, .2) {};

draw[red,use Hobby shortcut,text=black] (a.center) .. node[auto] {(1)} (b.center) .. node[auto] {(2)} (c.center) .. node[auto] {(3)} (d.center) .. node[auto] {(4)} (e.center);



end{tikzpicture}
end{document}