Adding PlotRangeClipping did not help, but adding PlotRange helped a lot

PlotRange -> {MinMax[Xticks], MinMax[Hticks]},

although when callouts were added the problem resurfaced again in the vertical direction above the plot. For now I've compromised by placing the first callout below and the remaining above.

All Above:

vs 1st Below:

X = 10000;
Xfractions = (#/X) & /@ Range[0, X];
Xcallout = Nearest[Xfractions, #/12][[1]] & /@ {1, 3, 5, 7, 9, 11};
brightBandsXRGBcolors = ColorData["BrightBands"][#] & /@ Xfractions;
brightBandsXHSIcolors = 
  ColorConvert[#, "HSB"] & /@ brightBandsXRGBcolors;
brightBandsXHSIcoords = List @@@ brightBandsXHSIcolors;
brightBandsXH = If[
     MemberQ[Xcallout, #[[1]]],
     Callout[#, Hue[#[[2]]], 
      Which[#[[1]] == Xcallout[[1]], Below, True, Above], 
      LeaderSize -> 0],
     #
     ] & /@ Thread[List[Xfractions, brightBandsXHSIcoords[[;; , 1]]]];
Xticks = (#/6) & /@ Range[0, 6];
Hticks = {0., 0.1, 0.2, 0.4, 0.6, 0.8, 1.};
plotTitle = 
  Column[{Style[
     "ColorData[\"BrightBands\"]:  parameters = " <> 
      ToString[ColorData["BrightBands", "ParameterCount"]] <> 
      ", parameter range = " <> 
      ToString[ColorData["BrightBands", "Range"]], {Black, 12}], 
    ColorData["BrightBands", "Image"]}, Center, 1];
LPXH = ListPlot[
   brightBandsXH,
   AxesOrigin -> {0, 0.},
   PlotRange -> {MinMax[Xticks], MinMax[Hticks]},
   PlotRangeClipping -> True,
   PlotStyle -> {Black, PointSize[0.0045]},
   Joined -> False,
   Ticks -> {Xticks, Hticks},
   GridLines -> {Xticks, Hticks},
   GridLinesStyle -> Automatic,
   AxesStyle -> {{Black, 11}, {Black, 11}},
   AxesLabel -> {Column[{Style["parameter", {Black, 13}], 
       Style["value", {Black, 12}]}, Left, 0], 
     Style["Hue", {Black, 12}]},
   PlotLabel -> plotTitle,
   AspectRatio -> 1/2,
   ImageSize -> {512, 512*2/3}
   ];
Print[LPXH];

Thank you Hans, this was the approach I needed. After adding some padding to the top, a blank line to the title, and repositioning the callout to Above, I received exactly the plot desired. :)

PlotRangePadding -> {{0., 0.}, {0., 0.05}},

Hah! Good catch. I likely misordered them in FrameTicks.